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Foundation of the minimal model

program

2014/4/16

version 0.01

Osamu Fujino

Department of Mathematics, Faculty of Science, KyotoUniversity, Kyoto 606-8502, Japan

E-mail address : fujino@math.kyoto-u.ac.jp

2010 Mathematics Subject Classification. Primary 14E30, 14F17;Secondary 14J05, 14E15

Abstract. We discuss various vanishing theorems. Then we es-tablish the fundamental theorems, that is, various Kodaira typevanishing theorems, the cone and contraction theorem, and so on,for quasi-log schemes.

Preface

This book is a completely revised version of the author’s unpub-lished manuscript:

• Osamu Fujino, Introduction to the minimal model program forlog canonical pairs, preprint 2008.

We note that the above unpublished manuscript is an expanded versionof the composition of

• Osamu Fujino, Vanishing and injectivity theorems for LMMP,preprint 2007

and

• Osamu Fujino, Notes on the log minimal model program, preprint2007.

We also note that this book is not an introductory text book of theminimal model program.

One of the main purposes of this book is to establish the funda-mental theorems, that is, various Kodaira type vanishing theorems,the cone and contraction theorem, and so on, for quasi-log schemes.The notion of quasi-log schemes was introduced by Florin Ambro inhis epoch-making paper:

• Florin Ambro, Quasi-log varieties, Tr. Mat. Inst. Steklova240 (2003), 220–239.

The theory of quasi-log schemes is extremely powerful. Unfortu-nately, it has not been popular yet because Ambro’s paper has severaldifficulties. Moreover, the author’s paper:

• Osame Fujino, Fundamental theorems for the log minimal modelprogram, Publ. Res. Inst. Math. Sci. 47 (2011), no. 3, 727–789

recovered the main result of Ambro’s paper, that is, the cone and con-traction theorem for normal pairs, without using the theory of quasi-logschemes. Note that the author’s approach in the above paper is suffi-cient for the fundamental theorems of the minimal model program forlog canonical pairs.

iii

iv PREFACE

Recently, the author proved that every quasi-projective semi logcanonical pair has a natural quasi-log structure which is compatiblewith the original semi log canonical structure in

• Osamu Fujino, Fundamental theorems for semi log canonicalpairs, Algebraic Geometry 1 (2014), no. 2, 194–228.

This result shows that the theory of quasi-log schemes is indispens-able for the study of semi log canonical pairs. Now the importanceof the theory of quasi-log schemes is increasing. In this book, we willestablish the foundation of quasi-log schemes.

One of the author’s main contributions in the above papers is tointroduce the theory of mixed Hodge structures on cohomology groupswith compact support to the minimal model program systematically.By pursuing this approach, we can naturally obtain a correct general-ization of the Fujita–Kawamata semipositivity theorem in

• Osamu Fujino, Taro Fujisawa, Variations of mixed Hodge struc-ture and semipositivity theorems, to appear in Publ. Res. Inst.Math. Sci.

This new powerful semipositivity theorem leads to the proof of theprojectivity of the coarse moduli spaces of stable varieties in

• Osamu Fujino, Semipositivity theorems for moduli problems,preprint 2012.

Note that a stable variety is a projective semi log canonical varietywith ample canonical divisor.

Anyway, the theory of quasi-log schemes seems to be indispensablefor the study of higher-dimensional algebraic varieties and its impor-tance is increasing now.

On page 57 in

• Janos Kollar, Shigefumi Mori, Birational geometry of alge-braic varieties, Cambridge University Press, 1998,

which is a standard text book on the minimal model program, theauthors wrote:

Log canonical: This is the largest class where discrep-ancy still makes sense. It contains many cases thatare rather complicated from the cohomological pointof view. Therefore it is very hard to work with.

On page 209, they also wrote:

The theory of these so-called semi-log canonical (slcfor short) pairs is not very much different from the lccase but it needs some foundational work.

PREFACE v

By the author’s series of papers including this book, we greatlyimprove the situation around log canonical pairs and semi log canonicalpairs from the cohomological point of view.

Acknowledgments. The author would like to thank ProfessorsShigefumi Mori, Yoichi Miyaoka, Noboru Nakayama, Daisuke Mat-sushita, and Hiraku Kawanoue, who were the members of the seminarswhen he was a graduate student at RIMS, Kyoto. In the seminarsat RIMS in 1997, he learned the foundation of the minimal modelprogram by reading a draft of [KoMo]. He thanks Professors TakaoFujita, Noboru Nakayama, Hiromichi Takagi, Florin Ambro, HiroshiSato, Takeshi Abe, Masayuki Kawakita, Yoshinori Gongyo, YoshinoriNamikawa, and Hiromu Tanaka for discussions, comments, and ques-tions. He also would like to thank Professor Janos Kollar for giving himmany comments on the preliminary version of this book and showinghim many examples. Finally, the author thanks Professors ShigefumiMori, Shigeyuki Kondo, Takeshi Abe, and Yukari Ito for warm encour-agement during the preparation of this book.

The author was partially supported by the Grant-in-Aid for YoungScientists (A) ]20684001 and ]24684002 from JSPS.

April 16, 2014Osamu Fujino

Contents

Preface iii

Guide for the reader xi

Chapter 1. Introduction 11.1. Mori’s cone and contraction theorem 11.2. What is a quasi-log scheme? 31.3. Motivation 51.4. Background 101.5. Comparison with the unpublished manuscript 111.6. Related papers 121.7. Notation and convention 13

Chapter 2. Preliminaries 152.1. Divisors, Q-divisors, and R-divisors 152.2. Kleiman–Mori cone 222.3. Singularities of pairs 242.4. Iitaka dimension, movable and pseudo-effective divisors 36

Chapter 3. Classical vanishing theorems and some applications 413.1. Kodaira vanishing theorem 423.2. Kawamata–Viehweg vanishing theorem 483.3. Viehweg vanishing theorem 553.4. Nadel vanishing theorem 603.5. Miyaoka vanishing theorem 613.6. Kollar injectivity theorem 633.7. Enoki injectivity theorem 643.8. Fujita vanishing theorem 683.9. Applications of Fujita vanishing theorem 763.10. Tanaka vanishing theorems 793.11. Ambro vanishing theorem 803.12. Kovacs’s characterization of rational singularities 823.13. Basic properties of dlt pairs 843.14. Elkik–Fujita vanishing theorem 913.15. Method of two spectral sequences 95

vii

viii CONTENTS

3.16. Toward new vanishing theorems 98

Chapter 4. Minimal model program 1034.1. Fundamental theorems for klt pairs 1034.2. X-method 1054.3. MMP for Q-factorial dlt pairs 1074.4. BCHM and some related results 1124.5. Fundamental theorems for normal pairs 1204.6. Lengths of extremal rays 1264.7. Shokurov polytope 1304.8. MMP for lc pairs 1374.9. Non-Q-factorial MMP 1454.10. MMP for log surfaces 1484.11. On semi log canonical pairs 153

Chapter 5. Injectivity and vanishing theorems 1575.1. Main results 1575.2. Simple normal crossing pairs 1605.3. Du Bois complexes and Du Bois pairs 1655.4. Hodge theoretic injectivity theorems 1695.5. Relative Hodge theoretic injectivity theorem 1745.6. Injectivity, vanishing, and torsion-free theorems 1765.7. Vanishing theorems of Reid–Fukuda type 1825.8. From SNC pairs to NC pairs 1885.9. Examples 193

Chapter 6. Fundamental theorems for quasi-log schemes 2016.1. Overview 2016.2. On quasi-log schemes 2036.3. Basic properties of quasi-log schemes 2066.4. On quasi-log structures of normal pairs 2156.5. Basepoint-free theorem for quasi-log schemes 2176.6. Rationality theorem for quasi-log schemes 2216.7. Cone theorem for quasi-log schemes 2266.8. On quasi-log Fano schemes 2326.9. Basepoint-free theorem of Reid–Fukuda type 233

Chapter 7. Some supplementary topics 2397.1. Alexeev’s criterion for S3 condition 2397.2. Cone singularities 2477.3. Francia’s flip revisited 2517.4. A sample computation of a log flip 2537.5. A non-Q-factorial flip 256

CONTENTS ix

Bibliography 259

Index 271

Guide for the reader

In Chapter 1, we start with Mori’s cone theorem for smooth pro-jective varieties and his contraction theorem for smooth threefolds. Itis one of the starting points of the minimal model program. So theminimal model program is sometimes called Mori’s program. We alsoexplain some examples of quasi-log schemes, the motivation of our van-ishing theorems, the background of this book, the author’s related pa-pers, and so on, for the reader’s convenience. Chapter 2 collects severaldefinitions and preliminary results. Almost all the topics in this chap-ter are well known to the experts and are indispensable for the study ofthe minimal model program. We recommend the reader to be familiarwith them. In Chapter 3, we discuss various Kodaira type vanishingtheorems and several applications. Although this chapter contains sev-eral new results and arguments, almost all the results are standardand are known to the experts. Chapter 4 is a survey on the minimalmodel program. We discuss the basic results of the minimal modelprogram, the recent results by Birkar–Cascini–Hacon–McKernan, andvarious results on log canonical pairs, log surfaces, semi log canonicalpairs by the author, and so on, without proof. Chapter 5 is devoted tothe injectivity, vanishing, and torsion-free theorems for reducible vari-eties. They are generalizations of Kollar’s corresponding results fromthe mixed Hodge theoretic viewpoint and play crucial roles in the the-ory of quasi-log schemes. Chapter 6 is the main part of this book. Weprove the adjunction and the vanishing theorem for quasi-log schemesas applications of the results in Chapter 5. Then we establish thebasepoint-free theorem, the rationality theorem, and the cone theoremfor quasi-log schemes, and so on. Chapter 7 collects some supplemen-tary results and examples. We recommend the reader who is familiarwith the traditional minimal model program and is only interested inthe theory of quasi-log schemes to go directly to Chapter 6.

xi

CHAPTER 1

Introduction

The minimal model program is sometimes called Mori’s programor Mori theory. This is because Shigefumi Mori’s epoch-making pa-per [Mo2] is one of the starting points of the minimal model program.Therefore, we quickly review Mori’s results in [Mo1] and [Mo2] in Sec-tion 1.1. In Section 1.2, we explain some basic examples of quasi-logschemes. By using the theory of quasi-log schemes, we can treat logcanonical pairs, non-klt loci of log canonical pairs, semi log canonicalpairs, and so on, on an equal footing. By [F33], the theory of quasi-logschemes seems to be indispensable for the study of semi log canonicalpairs. In Section 1.3, we explain some vanishing theorems, which aremuch sharper than the usual Kawamata–Viehweg vanishing theoremand the algebraic version of the Nadel vanishing theorem, in order tomotivate the reader to read this book. In Section 1.4, we give sev-eral historical comments on this book and the recent developments ofthe minimal model program for the reader’s convenience. We explainthe reason of the delay of the publication of this book. In Section1.5, we compare this book with the unpublished manuscript writtenand circulated in 2008. In Section 1.6, we quickly review the author’srelated papers and results for the reader’s convenience. In the finalsection: Section 1.7, we fix the notation and some conventions of thisbook.

1.1. Mori’s cone and contraction theorem

In his epoch-making paper [Mo2], Shigefumi Mori obtained thecone and contraction theorem. It is one of the starting points of Mori’sprogram or the minimal model program (MMP, for short).

Theorem 1.1.1 (Cone theorem). Let X be a smooth projective va-riety defined over an algebraically closed field. Then we have the fol-lowing properties.

(i) There are at most countably many (possibly singular) rationalcurves Ci on X such that

0 < −(Ci ·KX) ≤ dimX + 1,

1

2 1. INTRODUCTION

and

NE(X) = NE(X)KX≥0 +∑

R≥0[Ci].

Note that NE(X) is the Kleiman–Mori cone of X, that is,the closed convex cone spanned by the numerical equivalenceclasses of effective 1-cycles on X.

(ii) For any positive number ε and any ample Cartier divisor Hon X, we have

NE(X) = NE(X)(KX+εH)≥0 +∑finite

R≥0[Ci].

The proof of Theorem 1.1.1 in [Mo2] depends on Mori’s bend andbreak technique (see, for example, [KoMo, Chapter 1]). It was in-vented in [Mo1] to prove the Hartshorne conjecture.

Theorem 1.1.2 (Hartshorne conjecture, see [Mo1]). Let X be ann-dimensional smooth projective variety defined over an algebraicallyclosed field. If the tangent bundle TX is an ample vector bundle, thenX is isomorphic to Pn.

Note that Theorem 1.1.1 contains the following highly nontrivialtheorem.

Theorem 1.1.3 (Existence of rational curves). Let X be a smoothprojective variety defined over an algebraically closed field. If KX isnot nef, that is, there exists an irreducible curve C on X such thatKX · C < 0, then X contains a (possibly singular) rational curve.

There is no known proof of Theorem 1.1.3 which does not use pos-itive characteristic techniques even when the characteristic of the basefield is zero.

In [Mo2], Shigefumi Mori obtained the contraction theorem forsmooth projective threefold defined over C.

Theorem 1.1.4 (Contraction theorem, see [KoMo, Theorem 1.32]).Let X be a smooth projective threefold defined over C. Let R be anyKX-negative extremal ray of NE(X). Then there is a contraction mor-phism ϕR : X → Y associated to R.

The following is a list of all possibilities for ϕR.

E: (Exceptional). dimY = 3, ϕR is birational and there are fivetypes of local behavior near the contracted surfaces.E1: ϕR is the (inverse of the) blow-up of a smooth curve in

the smooth projective threefold Y .E2: ϕR is the (inverse of the) blow-up of a smooth point of the

smooth projective threefold Y .

1.2. WHAT IS A QUASI-LOG SCHEME? 3

E3: ϕR is the (inverse of the) blow-up of an ordinary doublepoint of Y . Note that an ordinary double point is locallyanalytically given by the equation

x2 + y2 + z2 + w2 = 0.

E4: ϕR is the (inverse of the) blow-up of a point of Y whichis locally analytically given by the equation

x2 + y2 + z2 + w3 = 0.

E5: ϕR contracts a smooth P2 with normal bundle OP2(−2) toa point of multiplicity 4 on Y which is locally analyticallythe quotient of C3 by the involution

(x, y, z) 7→ (−x,−y,−z).

C: (Conic bundle). dimY = 2 and ϕR is a fibration whose fibersare plane conics. Of course, general fibers are smooth.

D: (Del Pezzo fibration). dimY = 1 and general fibers of ϕR areDel Pezzo surfaces.

F: (Fano variety). dimY = 0, −KX is ample. Therefore, X is asmooth Fano threefold with the Picard number ρ(X) = 1.

For Mori’s bend and break technique, see, for example, [Ko7],[Deb], and [KoMo, Chapter 1]. For the details of the results in thissection, see the original papers [Mo1] and [Mo2]. We also recommendthe reader to see a good survey [Mo6]. After the epoch-making paper[Mo2], Shigefumi Mori classified three-dimensional terminal singulari-ties in [Mo3] (see also [R2]) and then established the flip theorem forterminal threefolds in [Mo5]. By these results with the works of Reid,Kawamata, Shokurov, and others, we obtained the existence theoremof minimal models for Q-factorial terminal threefolds.

Note that a shortest way to prove the existence of minimal modelsfor threefolds is now the combination of Shokurov’s proof of 3-fold plflips described in [Cor] and the reduction theorem explained in [F13].By this method, we are released from Mori’s deep classification of three-dimensional terminal singularities.

One of the main purposes of this book is to establish the coneand contraction theorem for quasi-log schemes, that is, the cone andcontraction theorem for highly singular schemes.

1.2. What is a quasi-log scheme?

In this section, we informally explain why it is natural to considerquasi-log schemes (see Section 6.4).

4 1. INTRODUCTION

Let (Z,BZ) be a log canonical pair and let f : V → Z be a resolu-tion with

KV + S +B = f ∗(KZ +BZ),

where Supp(S + B) is a simple normal crossing divisor, S is reduced,and bBc ≤ 0. It is very important to consider the non-klt locus Wof the pair (Z,BZ), that is, W = f(S). We consider the short exactsequence:

0→ OV (−S + d−Be)→ OV (d−Be)→ OS(d−Be)→ 0.

We put KS + BS = (KV + S + B)|S. In our case, BS = B|S. By theKawamata–Viehweg vanishing theorem, we have

Rif∗OV (−S + d−Be) = 0

for every i > 0. Since d−Be is effective and f -exceptional, we havef∗OV (d−Be) ' OZ . Therefore, we obtain the following short exactsequence:

0→ f∗OV (−S + d−Be)→ OZ → f∗OS(d−BSe)→ 0.

This implies

OW ' f∗OS(d−BSe).Note that the ideal sheaf f∗OV (−S + d−Be) is denoted by J (Z,BZ)and is called the multiplier ideal sheaf of the pair (Z,BZ).

Therefore, it is natural to introduce the following notion. Preciselyspeaking, a qlc pair is a quasi-log scheme with only qlc singularities.

Definition 1.2.1 (Qlc pairs). A qlc pair [X,ω] is a scheme X en-dowed with an R-Cartier divisor (or R-line bundle) ω such that thereis a proper morphism f : (Y,BY ) → X satisfying the following condi-tions.

(1) Y is a simple normal crossing divisor on a smooth variety Mand there exists an R-divisor D on M such that Supp(D+Y )is a simple normal crossing divisor, Y and D have no commonirreducible components, and BY = D|Y .

(2) f∗ω ∼R KY +BY .(3) BY is a subboundary R-divisor, that is, bi ≤ 1 for every i when

BY =∑biBi.

(4) OX ' f∗OY (d−(B<1Y )e), where B<1

Y =∑

bi<1 biBi.

It is easy to see that the pair [W,ω], where ω = (KX +B)|W , withf : (S,BS) → W satisfies the definition of qlc pairs. We note thatthe pair [Z,KZ + BZ ] with f : (V, S + B)→ Z is also a qlc pair sincef∗OV (d−Be) ' OZ . Thus, we can treat log canonical pairs and non-klt

1.3. MOTIVATION 5

loci of log canonical pairs in the same framework once we introduce thenotion of qlc pairs.

Moreover, we have:

Theorem 1.2.2. Let (X,∆) be a quasi-projective semi log canonicalpair. Then [X,KX + ∆] is naturally a qlc pair.

Theorem 1.2.2 is the main theorem of [F33], which is highly non-trivial and depends on the recent development of the theory of partialresolution of singularities for reducible varieties (see [BM] and [BVP]).For the details of Theorem 1.2.2, see [F33] (see also Theorem 4.11.9below).

Anyway, by Theorem 1.2.2, we can treat log canonical pairs, non-kltloci of log canonical pairs, quasi-projective semi log canonical pairs, andso on, on an equal footing by using the theory of quasi-log schemes. Theauthor thinks that Theorem 1.2.2 drastically increased the importanceof the theory of quasi-log schemes.

In this book, we establish the fundamental theorems, that is, variousKodaira type vanishing theorems, the cone and contraction theorem,and so on, for quasi-log schemes. For that purpose, we prove the Hodgetheoretic injectivity theorem for simple normal crossing pairs (see The-orem 5.1.1) and the injectivity, vanishing, and torsion-free theoremsfor simple normal crossing pairs (see Theorem 5.1.3). The main in-gredient of our framework is the theory of mixed Hodge structures oncohomology with compact support.

1.3. Motivation

The following results will motivate the reader to study our newframework, which is more powerful than the traditional X-methodbased on the Kawamata–Viehweg vanishing theorem (see, for example,[KMM] and [KoMo]), and the theory of algebraic multiplier idealsheaves (see, for example, [La2, Part Three]), which depends on theNadel vanishing theorem.

Theorem 1.3.1. Let X be a normal projective variety and let B bean effective R-divisor on X such that (X,B) is log canonical. Let Lbe a Cartier divisor on X. Assume that L − (KX + B) is ample. LetCi be any set of log canonical centers of the pair (X,B). We putW =

∪Ci with the reduced scheme structure. Then we have

H i(X, IW ⊗OX(L)) = 0

6 1. INTRODUCTION

for every i > 0, where IW is the defining ideal sheaf of W on X. Inparticular, the natural restriction map

H0(X,OX(L))→ H0(W,OW (L))

is surjective. Therefore, if (X,B) has a zero-dimensional log canonicalcenter, then the linear system |L| is not empty and the base locus of|L| contains no zero-dimensional log canonical centers of (X,B).

More generally, we have:

Theorem 1.3.2. Let X be a normal projective variety and let B bean effective R-divisor on X such that KX + B is R-Cartier. Let L bea Cartier divisor on X. Assume that L− (KX +B) is nef and log bigwith respect to the pair (X,B). Let Nlc(X,B) denote the non-lc locusof the pair (X,B). Let Ci be any set of log canonical centers of thepair (X,B). We put

W = Nlc(X,B) ∪∪

Ci.

Then W has a natural scheme structure induced by the pair (X,B),and

H i(X, IW ⊗OX(L)) = 0

holds for every i > 0, where IW is the defining ideal sheaf of W on X.

Although we did not define the scheme structure of W explicitlyhere, it is natural and Theorem 1.3.2 is a generalization of Theorem1.3.1. Note that Theorem 1.3.2 is a very special case of Theorem 6.3.4.We also note that the proof of Theorem 1.3.2 is much harder than theproof of Theorem 1.3.1.

1.3.3. In Theorem 1.3.2, if we assume that W is the union of allthe log canonical centers of (X,B), then IW becomes the multiplierideal sheaf J (X,B) of the pair (X,B). In this case, W is the non-kltlocus of the pair (X,B) and the vanishing theorem in Theorem 1.3.2is nothing but the Nadel vanishing theorem:

H i(X,J (X,B)⊗OX(L)) = 0

for every i > 0. Therefore, Theorem 1.3.2 is a generalization of theNadel vanishing theorem. It is obvious that Theorem 1.3.2 is alsoa generalization of the Kawamata–Viehweg vanishing theorem. Notethat IW = OX when W and Nlc(X,B) are empty.

Let us see a simple setting to understand the difference betweenour new framework and the traditional one.

1.3. MOTIVATION 7

1.3.4. Let X be a smooth projective surface and let C1 and C2 besmooth curves on X. Assume that C1 and C2 intersect only at a pointP transversally. Let L be a Cartier divisor onX such that L−(KX+B)is ample, where B = C1 +C2. It is obvious that (X,B) is log canonicaland P is a log canonical center of (X,B). Then, by Theorem 1.3.1, wecan directly obtain

H i(X, IP ⊗OX(L)) = 0

for every i > 0, where IP is the defining ideal sheaf of P on X.In the classical framework, we prove it as follows. Let C be a general

curve passing through P . We take small positive rational numbers εand δ such that (X, (1 − ε)B + δC) is log canonical and is kawamatalog terminal outside P and that P is an isolated log canonical centerof (X, (1− ε)B + δC). Since ε and δ are small,

L− (KX + (1− ε)B + δC)

is still ample. By the Nadel vanishing theorem, we obtain

H i(X, IP ⊗OX(L)) = 0

for every i > 0. We note that IP is nothing but the multiplier idealsheaf associated to the pair (X, (1− ε)B + δC).

By our new vanishing theorems (see, Theorem 1.3.1, Theorem 1.3.2,and so on), the reader will be released from annoyance of perturbingcoefficients of boundary divisors.

In Chapter 5, we will generalize Kollar’s torsion-free and vanishingtheorem (see Theorem 5.1.3). As an application, we will prove Theorem6.3.4, which contains Theorem 1.3.2. Note that Kollar’s torsion-freeand vanishing theorem is equivalent to Kollar’s injectivity theorem.

Let us try to give a proof of a very special case of Theorem 1.3.1by using Kollar’s torsion-free and vanishing theorem.

Theorem 1.3.5. Let S be a normal projective surface which hasonly one simple elliptic Gorenstein singularity Q ∈ S. We put X =S × P1 and B = S × 0. Then the pair (X,B) is log canonical. It iseasy to see that P = (Q, 0) ∈ X is a log canonical center of (X,B). LetL be a Cartier divisor on X such that L − (KX + B) is ample. Thenwe have

H i(X, IP ⊗OX(L)) = 0

for every i > 0, where IP is the defining ideal sheaf of P on X. Wenote that X is not kawamata log terminal and that P is not an isolatedlog canonical center of (X,B).

8 1. INTRODUCTION

Proof. Let ϕ : T → S be the minimal resolution. Then we canwrite KT + C = ϕ∗KS, where C is the ϕ-exceptional elliptic curve onT . We put Y = T×P1 and f = ϕ×idP1 : Y → X, where idP1 : P1 → P1

is the identity. Then f is a resolution of X and we can write

KY +BY + E = f ∗(KX +B),

where BY is the strict transform of B on Y and E ' C × P1 is theexceptional divisor of f . Let g : Z → Y be the blow-up along E ∩BY .Then we can write

KZ +BZ + EZ + F = g∗(KY +BY + E) = h∗(KX +B),

where h = f g, BZ (resp. EZ) is the strict transform of BY (resp. E)on Z, and F is the g-exceptional divisor. We note that

IP ' h∗OZ(−F ) ⊂ h∗OZ ' OX .

Since −F = KZ +BZ + EZ − h∗(KX +B), we have

IP ⊗OX(L) ' h∗OZ(KZ +BZ + EZ)⊗OX(L− (KX +B)).

So, it is sufficient to prove that

H i(X, h∗OZ(KZ +BZ + EZ)⊗ L) = 0

for every i > 0 and any ample line bundle L on X. We consider theshort exact sequence

0→ OZ(KZ)→ OZ(KZ + EZ)→ OEZ(KEZ

)→ 0.

We can easily check that

0→ h∗OZ(KZ)→ h∗OZ(KZ + EZ)→ h∗OEZ(KEZ

)→ 0

is exact and

Rih∗OZ(KZ + EZ) ' Rih∗OEZ(KEZ

)

for every i > 0 because Rih∗OZ(KZ) = 0 for every i > 0. The factRih∗OZ(KZ) = 0 for every i > 0 is a special case of Kollar’s torsion-freetheorem since h is birational. We can directly check that

R1h∗OEZ(KEZ

) ' R1f∗OE(KE) ' OD(KD),

where D = Q× P1 ⊂ X. Therefore, R1h∗OZ(KZ + EZ) ' OD(KD) isa torsion sheaf on X. However, it is torsion-free as a sheaf on D. It isa generalization of Kollar’s torsion-free theorem. We consider

0→ OZ(KZ + EZ)→ OZ(KZ +BZ + EZ)→ OBZ(KBZ

)→ 0.

1.3. MOTIVATION 9

We note that BZ ∩ EZ = ∅. Thus, we have

0→ h∗OZ(KZ + EZ)→ h∗OZ(KZ +BZ + EZ)→ h∗OBZ(KBZ

)

δ→ R1h∗OZ(KZ + EZ)→ · · · .

Since Supph∗OBZ(KBZ

) = B, δ is a zero map by R1h∗OZ(KZ +BZ) 'OD(KD). Therefore, we know that the following sequence

0→ h∗OZ(KZ + EZ)→ h∗OZ(KZ +BZ + EZ)→ h∗OBZ(KBZ

)→ 0

is exact. By Kollar’s vanishing theorem on BZ , it is sufficient to provethat H i(X, h∗OZ(KZ + EZ) ⊗ L) = 0 for every i > 0 and any ampleline bundle L. We have

H i(X, h∗OZ(KZ)⊗ L) = H i(X, h∗OEZ(KEZ

)⊗ L) = 0

for every i > 0 by Kollar’s vanishing theorem. By the following exactsequence

· · · → H i(X, h∗OZ(KZ)⊗ L)→ H i(X, h∗OZ(KZ + EZ)⊗ L)

→ H i(X, h∗OEZ(KEZ

)⊗ L)→ · · · ,

we obtain the desired vanishing theorem. Anyway, we have

H i(X, IP ⊗OX(L)) = 0

for every i > 0.

The actual proof of Theorem 1.3.2 (see Theorem 6.3.4) dependson much more sophisticated arguments of the theory of mixed Hodgestructures on cohomology groups with compact support.

Remark 1.3.6. In Theorem 1.3.5, X is log canonical and is notkawamata log terminal. Note that D = Q × P1 ⊂ X is a one-dimensional log canonical center of X passing through P . Therefore,in order to prove Theorem 1.3.5, we can not apply the traditional per-turbation technique as in 1.3.4.

In Chapter 5, we will first generalize Kollar’s injectivity theorem(see Theorem 5.1.1 and Theorem 5.1.2). Next, we will obtain a general-ization of Kollar’s torsion-free and vanishing theorem as an application(see Theorem 5.1.3). Finally, we will apply it to quasi-log schemes inChapter 6.

10 1. INTRODUCTION

1.4. Background

In this section, we give some historical comments on the theory ofquasi-log schemes and the recent developments of the minimal modelprogram.

In November 2001, Ambro’s preprint:

• Florin Ambro, Generalized log varieties

appeared on the archive. It was a preprint version of [Am1]. I thinkthat it did not attract so much attention when it appeared on thearchive. A preprint version of [Sh4], which was first circulated around2000, attracted much more attention than Ambro’s preprint. In Feb-ruary 2002, a working seminar on Shokurov’s preprint, which was or-ganized by Alessio Corti, started in the Newton Institute. I stayed atthe Newton Institute in February and March to attend the workingseminar. The book [Cored] is an outcome of this working seminar.

On October 5, 2006, a preprint version of [BCHM] appeared onthe archive. In November, I invited Hiromichi Takagi to Nagoya fromTokyo and tried to understand the preprint. Although it was muchmore complicated than the published version, we soon recognized thatit is essentially correct. This meant that I lost my goal in life. InDecember 2006, Christopher Hacon and James McKernan gave talkson [BCHM] at Echigo Yuzawa in Japan. In January 2007, HiromichiTakagi gave a series of lectures on [BCHM] for graduate students inKyoto. I visited Kyoto to attend his lectures. If I remember cor-rectly, Masayuki Kawakita had already understood [BCHM] in Jan-uary 2007. In March 2007, Caucher Birkar visited Japan and gaveseveral talks on his results in Tokyo and Kyoto. In Japan, a preprintversion of [BCHM] was digested quickly. We note that HiromichiTakagi, Masayuki Kawakita, and I were the participants of the work-ing seminar on Shokurov’s preprint ([Sh4]) in the Newton Institute in2002. After I read a preprint version of [BCHM], I decided to establishvanishing theorems sufficient for the theory of quasi-log schemes. Wehad already known that Ambro’s paper [Am1] contains various diffi-culties. In April 2007, I finished a preprint version of [F14] and sent itto some experts. Then I visited MSRI to attend a workshop. The titleof the workshop is Hot topics: Minimal and Canonical Models in Al-gebraic Geometry. Of course, I tried to publish [F14]. Unfortunately,the referees did not understand the importance of [F14]. I think thatmany experts including the referees were busy in reading [BCHM] andwere not interested in [F14] in 2007. So I changed my plan and decidedto combine [F14] and [F15] and publish it as a book. In June 2008,I sent a preliminary version of [F17], which is version 2.0, to some

1.5. COMPARISON WITH THE UNPUBLISHED MANUSCRIPT 11

experts including Janos Kollar. He kindly gave me some commentsalthough I did not understand them. After I moved to Kyoto fromNagoya in October, I visited Princeton to ask advice to Janos Kollarin November. When I visited Princeton, he was preparing [KoKo] andgave me a copy of a draft. During my stay at Princeton, he askedChristopher Hacon about the existence of dlt blow-ups by e-mail. Hegave me a copy of the e-mail from Hacon which proved the existence ofdlt blow-ups. In December 2008, I suddenly came up with a good ideawhen I attended Professor Hironaka’s talk at RIMS on the resolution ofsingularities. Then I soon got a very short proof of the basepoint-freetheorem for log canonical pairs without using the theory of quasi-logschemes (see [F27]). By using dlt blow-ups, I succeeded in proving thefundamental theorems for log canonical pairs very easily (see [F27]).In [F28], I recovered the main result of [Am1], that is, the fundamen-tal theorems for normal pairs, and got some generalizations withoutusing the theory of quasi-log schemes. Therefore, I lost much of myinterests in the theory of quasi-log schemes. This is the main reason ofthe delay of the revision and publication of [F17]. In May 2011, JanosKollar informed me of the development of the theory of partial reso-lution of singularities for reducible varieties in Kyoto. It looked veryattractive for me. In September, a preprint version of [BVP] appearedon the archive. By using this new result, in January 2012, I proved thatevery quasi-projective semi log canonical pair has a natural quasi-logstructure with only quasi-log canonical singularities (see [F33]). Thisresult shows that the theory of quasi-log schemes is indispensable forthe cohomological study of semi log canonical pairs.

After I wrote [F17], the minimal model theory for log canonicalpairs has developed. For the details, see, for example, [Bir4], [F38],[FG1], [FG2], [HaX1], [HaX2], [HaMcX], [Ko13], and so on.

1.5. Comparison with the unpublished manuscript

In this section, we compare this book with the author’s unpublishedmanuscript:

• Osamu Fujino, Introduction to the minimal model program forlog canonical pairs, preprint 2008

for the reader’s convenience. The version 6.01 of the above manuscript(see [F17]), which was circulated in January 2009, is available fromarXiv.org. We think that [F17] has already been referred and used inmany papers.

This book does not cover Subsections 3.1.4, 3.2.6, and 3.2.7 in[F17]. Subsection 3.1.4 in [F17] is included in [F38, Section 7] with

12 1. INTRODUCTION

some revisions. Subsections 3.2.6 and 3.2.7 in [F17] are essentiallycontained in [F39]. For the details, see [F38] and [F39].

Chapter 2 of [F17] is now the main part of Chapter 5 in this book.Note that we greatly revised the proof of the Hodge theoretic injectivitytheorems, which were called the fundamental injectivity theorems in[F17]. Please compare [F17, Section 2.3] with Section 5.4. Note thatthe results in Chapter 5 are better than those in Chapter 2 of [F17].

Chapter 6 of this book consists of Section 3.2 and Section 3.3 in[F17] and Section 4.1 in [F17] with several revisions. Of course, thequality of Chapter 6 of this book is much better than that of the cor-responding part of [F17].

Chapter 3 except Section 3.13 and Section 3.15 is new. AlthoughChapter 3 contains some new arguments and some new results, almostall results are standard or known to the experts. We wrote Chapter 3for the reader’s convenience.

In this book, we expanded the explanation of the minimal modelprogram compared with [F17]. It is Chapter 4 of this book. Chap-ter 4 contains many results obtained after [F17] was written in 2008.We hope that Chapter 4 will help the reader understand the recentdevelopments of the minimal model program.

1.6. Related papers

In this section, we review the author’s related papers for the reader’sconvenience.

In [F6, Section 2], we obtained some special cases of the torsion-free theorem for log canonical pairs and Kollar type vanishing theoremfor log canonical pairs. The semipositivity theorem in [F6] is nowcompletely generalized in [FF] (see also [FFS]). The paper [FF] isin the same framework as [F32], [F36], and this book. Therefore,we recommend the reader to see [FF] after reading this book. Thepaper [F23] is a survey article of the theory of quasi-log schemes. Werecommend the reader to see [F23] before reading Chapter 6. The twoshort papers [F18] and [F27] are almost sufficient for the fundamentaltheorems for projective log canonical pairs although the paper [F28]superseded [F18] and [F27]. As a nontrivial application of [F28], weobtained the minimal model theory for Q-factorial surfaces in [F29](see Section 4.10). The results in [F29] are sharper than the traditionalminimal model theory for singular surfaces. In [F19] and [F21], wegeneralized the effective basepoint-free theorems for log canonical pairs.We can not reach these results by the traditional X-method and thetheory of multiplier ideal sheaves. Chapter 5 of this book contains the

1.7. NOTATION AND CONVENTION 13

main results of the papers [F32] and [F36]. However, this book doesnot contain applications discussed in [F32] and [F36]. In this book,we do not prove Theorem 1.2.2 (see also Section 4.11). For the detailson semi log canonical pairs, see [F33]. For applications to moduliproblems of stable varieties, see [F35].

In the author’s recent preprint [F39], we clarify the definition ofquasi-log structures and make the theory of quasi-log schemes moreflexible and more useful. Note that the definition of quasi-log schemesin this book is slightly different from Ambro’s original one althoughthey are equivalent. For the details of the relationship between ourdefinition and Ambro’s original one, see [F39]. In [F40], we introducevarious new operations for quasi-log structures. Then we prove thebasepoint-free theorem of Reid–Fukuda type for quasi-log schemes as anapplication (see Section 6.9). We note that the basepoint-free theoremof Reid–Fukuda type for quasi-log schemes was proved under someextra assumptions in [F17] and in this book (see Section 6.9).

1.7. Notation and convention

We fix the notation and the convention of this book.

1.7.1 (Schemes and varieties). A scheme means a separated schemeof finite type over an algebraically closed field k. A variety meansa reduced scheme, that is, a reduced separated scheme of finite typeover an algebraically closed field k. We note that a variety in thisbook may be reducible and is not always equidimensional. However,we sometimes implicitly assume that a variety is irreducible withoutmentioning it explicitly if there is no risk of confusion. If it is notexplicitly stated, then the field k is the complex number field C. Wenote that, by using the Lefschetz principle, we can extend almost allthe results over C in this book to the case when k is an arbitraryalgebraically closed field of characteristic zero.

1.7.2 (Birational map). A birational map f : X 99K Y betweenschemes means that f is a rational map such that there are Zariski

open dense subsets U of X and V of Y with f : U'−→ V .

1.7.3 (Exceptional locus). For a birational morphism f : X → Y ,the exceptional locus Exc(f) ⊂ X is the set

x ∈ X | f is not biregular at x,that is, the set of points x ∈ X where f−1 is not a morphism atf(x). We usually see Exc(f) as a subscheme with the induced reducedstructure.

14 1. INTRODUCTION

1.7.4 (Pairs). A pair [X,ω] consists of a scheme X and an R-Cartierdivisor (or R-line bundle) on X.

1.7.5 (Dualizing complex and dualizing sheaf). The symbol ω•X de-notes the dualizing complex of X. When X is an equidimensional va-riety with dimX = d, then we put ωX = H−d(ω•X) and call it thedualizing sheaf of X.

1.7.6 (see [KoMo, Definition 2.24]). Let X be an equidimensionalvariety, let f : Y → X be a (not necessarily proper) birational mor-phism from a normal variety Y , and let E be a prime divisor on Y .Any such E is called a divisor over X. The closure of f(E) ⊂ X iscalled the center of E on X.

1.7.7 (... for every m 0). The expression ‘... for every m 0’means that ‘there exists a positive number m0 such that ... for everym ≥ m0.’

1.7.8 (Z, Z≥0, Z>0, Q, R, R≥0, and R>0). The set of integers(resp. rational numbers or real numbers) is denoted by Z (resp. Q orR). The set of non-negative (resp. positive) real numbers is denoted byR≥0 (resp. R>0). Of course, Z≥0 (resp. Z>0) is the set of non-negative(resp. positive) integers.

CHAPTER 2

Preliminaries

In this chapter, we collect the basic definitions of the minimal modelprogram for the reader’s convenience.

In Section 2.1, we recall some basic definitions and properties ofQ-divisors and R-divisors. The use of R-divisors is indispensable forthe recent developments of the minimal model program. Moreover, wehave to treat R-divisors on reducible non-normal varieties in this book.In Section 2.2, we recall some basic definitions and properties of theKleiman–Mori cone. Note that Kleiman’s famous ampleness criteriondoes not always hold for complete non-projective singular algebraic va-rieties. In Section 2.3, we discuss discrepancy coefficients, singularitiesof pairs, negativity lemmas, and so on. They are very important inthe minimal model theory. In Section 2.4, we recall the Iitaka dimen-sion, the numerical Iitaka dimension, movable divisors, pseudo-effectivedivisors, Nakayama’s numerical dimension, and so on.

2.1. Divisors, Q-divisors, and R-divisors

Let us start with the definition of simple normal crossing divisorsand normal crossing divisors.

Definition 2.1.1 (Simple normal crossing divisors and normalcrossing divisors). Let X be a smooth algebraic variety. A reducedeffective Cartier divisor D on X is said to be a simple normal crossingdivisor (resp. normal crossing divisor) if for each closed point p of X,a local defining equation f of D at p can be written as

f = z1 · · · zjpin OX,p (resp. OX,p), where z1, · · · , zjp is a part of a regular systemof parameters.

Note that the notion of Q-factoriality plays important roles in theminimal model program.

Definition 2.1.2 (Q-factoriality). A normal variety X is said tobe Q-factorial if every prime divisor D on X is Q-Cartier, that is, somenon-zero multiple of D is Cartier.

15

16 2. PRELIMINARIES

Example 2.1.3 shows that the notion of Q-factoriality is very subtle.

Example 2.1.3 (cf. [Ka3]). We consider

X = (x, y, z, w) ∈ C4 | xy + zw + z3 + w3 = 0.

Claim. The algebraic variety X is Q-factorial. More precisely, Xis factorial, that is,

R = C[x, y, z, w]/(xy + zw + z3 + w3)

is a UFD.

Proof of Claim. By Nagata’s lemma (see [Mum2, p. 196]), itis sufficient to see that x ·R is a prime ideal of R and R[1/x] is a UFD.It is an easy exercise.

Claim. Let Xan be the associated analytic space of X. Then Xan

is not analytically Q-factorial.

Proof of Claim. We consider a germ of Xan around the origin.Then Xan is local analytically isomorphic to (xy − uv = 0) ⊂ C4.Therefore, Xan is not Q-factorial since the two divisors (x = u = 0)and (y = v = 0) intersect at a single point. Note that two Q-Cartierdivisors must intersect each other in codimension one.

Lemma 2.1.4 is well known and is sometimes very useful. For otherproofs, see [Ka2, Proposition 5.8], [KoMo, Corollary 2.63], and so on.

Lemma 2.1.4. Let f : X → Y be a birational morphism betweennormal varieties. Assume that Y is Q-factorial. Then the exceptionallocus Exc(f) of f is of pure codimension one.

Proof. Let x ∈ Exc(f) be a point. Without loss of generality, wemay assume thatX is affine by replacingX with an affine neighborhoodof x. We assume that X ⊂ CN , with coordinates t1, · · · , tN , and thatg = f−1 is the map given by ti = gi for i = 1, · · · , N , with gi ∈ C(Y ).It is obvious that gi = g∗ti. We put y = f(x). Since f−1 = g is notregular at y, we may assume that g1 is not regular at y. By assumption,the divisor class group of OY,y is torsion. Therefore, we can write

gm1 =u

vfor some positive integer m and some relatively prime elements u, v ∈OY,y. Since g1 is not regular at y, we have v(y) = 0. Note that Y isnormal. Therefore, (u = v = 0) has codimension two in Y and

(f ∗u = f ∗v = 0) = (tm1 f∗v = f∗v = 0) ⊃ (f∗v = 0) 3 x

has codimension one at x.

2.1. DIVISORS, Q-DIVISORS, AND R-DIVISORS 17

As is well-known, the notion of Q-divisors and R-divisors is indis-pensable for the minimal model program.

Definition 2.1.5 (Q-Cartier divisors and R-Cartier divisors). AnR-Cartier (resp. Q-Cartier) divisor D on a scheme X is a finite R-linear(Q-linear) combination of Cartier divisors.

Let us recall the definition of ample R-divisors.

Definition 2.1.6 (Ample R-divisors). Let π : X → S be a mor-phism between schemes. An R-Cartier divisor D on a scheme X is saidto be π-ample if D is a finite R>0-linear combination of π-ample Cartierdivisors on X. We simply say that D is ample when S is a point.

We need various operations of Q-divisors and R-divisors in thisbook.

2.1.7 (Q-divisors and R-divisors). Let B1 and B2 be two R-Cartierdivisors on a scheme X. Then B1 is linearly (resp. Q-linearly, or R-linearly) equivalent to B2, denoted by B1 ∼ B2 (resp. B1 ∼Q B2, orB1 ∼R B2) if

B1 = B2 +k∑i=1

ri(fi)

such that fi ∈ Γ(X,K∗X) and ri ∈ Z (resp. ri ∈ Q, or ri ∈ R) for everyi. Here, KX is the sheaf of total quotient rings of OX and K∗X is thesheaf of invertible elements in the sheaf of rings KX . We note that(fi) is a principal Cartier divisor associated to fi, that is, the image offi by Γ(X,K∗X) → Γ(X,K∗X/O∗X), where O∗X is the sheaf of invertibleelements in OX .

Let f : X → Y be a morphism between schemes. If there is anR-Cartier divisor B on Y such that

B1 ∼R B2 + f∗B,

then B1 is said to be relatively R-linearly equivalent to B2. It is denotedby B1 ∼R,f B2 or B1 ∼R,Y B2.

When X is complete, B1 is numerically equivalent to B2, denotedby B1 ≡ B2, if B1 · C = B2 · C for every curve C on X (see also 2.2.1below).

LetD be a Q-divisor (resp. R-divisor) on an equidimensional varietyX, that is, D is a finite formal Q-linear (resp. R-linear) combination

D =∑i

diDi

18 2. PRELIMINARIES

of irreducible reduced subschemes Di of codimension one. We definethe round-up dDe =

∑iddieDi (resp. round-down bDc =

∑ibdicDi),

where every real number x, dxe (resp. bxc) is the integer defined byx ≤ dxe < x + 1 (resp. x − 1 < bxc ≤ x). The fractional part D ofD denotes D − bDc. We set

D<1 =∑di<1

diDi, D≥1 =∑di≥1

diDi, and D=1 =∑di=1

Di.

We can defineD≤1, D>1, and so on, analogously. We callD a boundary(resp. subboundary) R-divisor if 0 ≤ di ≤ 1 (resp. di ≤ 1) for every i.

2.1.8 (Big divisors). Let us collect some basic definitions and prop-erties of big divisors. For the details, [La1, Section 2.2], [Mo4], [Nak2,Chapter II. §3.d], [U, Chapter II], and so on. For the details of bigR-divisors on (not necessarily normal) irreducible varieties, see [F33,Appendix A. Big R-divisors].

Definition 2.1.9 (Big Cartier divisors). Let X be a normal com-plete irreducible variety and let D be a Cartier divisor on X. Then Dis big if one of the following equivalent conditions holds.

(1) maxm∈Z>0

dim Φ|mD|(X) = dimX, where Φ|mD| : X 99K PN is

the rational map associated to the linear system |mD| andΦ|mD|(X) is the image of Φ|mD|.

(2) There exist a rational number α and a positive integer m0 suchthat

αmdimX ≤ dimH0(X,OX(mm0D))

for every m 0.

It is well known that we can take m0 = 1 in the condition (2) (see, forexample, [La1, Corollary 2.1.38], [Nak2, Chapter II.3.17. Corollary],and so on).

For non-normal varieties, we need the following definition.

Definition 2.1.10 (Big Cartier divisors on non-normal varieties).Let X be a complete irreducible variety and let D be a Cartier divisoron X. Then D is big if ν∗D is big on Xν , where ν : Xν → X is thenormalization.

Before we define big R-divisors, let us recall the definition of bigQ-divisors.

Definition 2.1.11 (Big Q-divisors). Let X be a complete irre-ducible variety and let D be a Q-Cartier divisor on X. Then D is bigif mD is a big Cartier divisor for some positive integer m.

2.1. DIVISORS, Q-DIVISORS, AND R-DIVISORS 19

We note the following obvious lemma.

Lemma 2.1.12. Let f : W → V be a birational morphism betweennormal complete irreducible varieties and let D be a Q-Cartier divisoron V . Then D is big if and only if so is f ∗D.

Next, let us define big R-divisors.

Definition 2.1.13 (Big R-divisors on complete varieties). An R-Cartier divisor D on a complete irreducible variety X is big if it can bewritten in the form

D =∑i

aiDi

where each Di is a big Cartier divisor and ai is a positive real numberfor every i.

Remark 2.1.14. Definition 2.1.13 is compatible with Definition2.1.11. This means that if a Q-Cartier divisor D is big in the senseof Definition 2.1.13 then D is big in the sense of Definition 2.1.11. Forthe details, see [F33, Appendix A].

We can check the following proposition.

Proposition 2.1.15 (see [F33, Proposition A.14]). Let D be anR-Cartier divisor on a normal complete irreducible variety X. Thenthe following conditions are equivalent.

(1) D is big.(2) There exist a positive rational number α and a positive integer

m0 such that

αmdimX ≤ dimH0(X,OX(bmm0Dc))for every m 0.

Note that we do not assume that X is projective in Proposition2.1.15. We omit the proof of Proposition 2.1.15 here since we do notuse it explicitly in this book. For the proof, see [F33, PropositionA.14].

Definition 2.1.16 (Big R-divisors on complete reducible varieties).LetX be a complete reducible variety and letD be an R-Cartier divisoron X. Then D is big if D|Xi

is big for every irreducible component Xi

of X.

Definition 2.1.17 (Relative big R-divisors). Let π : X → S bea proper morphism from a variety X to a scheme S and let D bean R-Cartier divisor on X. Then D is called π-big or big over S ifthe restriction of D to the geometric generic fiber of every irreduciblecomponent of π(X) is big.

20 2. PRELIMINARIES

The following version of Kodaira’s lemma is suitable for our pur-poses.

Lemma 2.1.18 (Kodaira). Let X be a projective irreducible varietyand let D be a big Cartier divisor on X. Let H be any Cartier divisoron X. Then there exists a positive integer a such that

H0(X,OX(aD −H)) 6= 0.

Therefore, we can write aD ∼ H+E for some effective Cartier divisorE on X.

Proof. We put n = dimX, Let ν : Xν → X be the normalization.We consider the following short exact sequence

0→ OX → ν∗OXν → δ → 0.

Note that dim Supp δ < n. By taking ⊗OX(mD) and taking cohomol-ogy, we obtain

0→ H0(X,OX(mD))→ H0(Xν ,OXν (mν∗D))

→ H0(X, δ ⊗OX(mD))→ · · · .Since ν∗D is big by definition, there is a positive rational number α1

such that dimH0(Xν ,OXν (mν∗D)) ≥ α1mn for every m 0. Since

dim Supp δ < n, there is a positive rational number α2 such thatdimH0(X, δ⊗OX(mD)) ≤ α2m

n for every m 0. Therefore, there isa positive rational number α3 such that dimH0(X,OX(mD)) ≥ α3m

n

for every m 0. By adding a sufficiently ample divisor to H, wemay assume that H is a very ample effective divisor on X. Note thatthere is a positive rational number α4 such that dimH0(H,OH(mD)) ≤α4m

n−1 since dimH = n− 1. Then, by

0→ H0(X,OX(mD −H))→ H0(X,OX(mD))

→ H0(H,OH(mD))→ · · · ,

we obtain H0(X,OX(aD −H)) 6= 0 for some positive integer a. We will repeatedly use Kodaira’s lemma (Lemma 2.1.18) and its

variants throughout this book.

2.1.19 (Semi-ample divisors). Let us recall some basic properties ofsemi-ample divisors. In this book, we have to deal with semi-ampleR-divisors.

Definition 2.1.20 (Semi-ample R-divisors). Let π : X → S be amorphism between schemes. An R-Cartier divisor D on X is π-semi-ample if D ∼R

∑i aiDi, where Di is a π-semi-ample Cartier divisor on

2.1. DIVISORS, Q-DIVISORS, AND R-DIVISORS 21

X and ai is a positive real number for every i. We simply say that Dis semi-ample when S is a point.

Remark 2.1.21. In Definition 2.1.20, we can replace D ∼R∑

i aiDi

with D =∑

i aiDi since every principal Cartier divisor on X is π-semi-ample.

We note the following two lemmas.

Lemma 2.1.22 (see [F28, Lemma 4.13]). Let D be an R-Cartierdivisor on X and let π : X → S be a morphism between schemes.Then the following conditions are equivalent.

(1) D is π-semi-ample.(2) There exists a morphism f : X → Y over S such that D ∼R

f∗A, where A is an R-Cartier divisor on Y which is ampleover S.

Proof. It is obvious that (1) follows from (2). If D is π-semi-ample, then we can write D ∼R

∑i aiDi as in Definition 2.1.20. By

replacing Di with its multiple, we may assume that π∗π∗OX(Di) →OX(Di) is surjective for every i. Let f : X → Y be a morphism overS obtained by the surjection π∗π∗OX(

∑iDi) → OX(

∑iDi). Then it

is easy to see that f : Y → X has the desired property.

Lemma 2.1.23 (see [F28, Lemma 4.14]). Let D be a Cartier divisoron X and let π : X → S be a morphism between schemes. If D is π-semi-ample in the sense of Definition 2.1.20, then D is π-semi-amplein the usual sense, that is, π∗π∗OX(mD)→ OX(mD) is surjective forsome positive integer m. This means that Definition 2.1.20 is compat-ible with the usual definition.

Proof. We write D ∼R∑

i aiDi as in Definition 2.1.20. Letf : X → Y be a morphism in Lemma 2.1.22 (2). By taking theStein factorization, we may assume that f has connected fibers. Byconstruction, Di ∼Q,f 0 for every i. By replacing Di with its multiple,we may assume that Di ∼ f ∗D′i for some Cartier divisor D′i on Y forevery i. Let U be any Zariski open set of Y on which D′i ∼ 0 for everyi. On f−1(U), we have D ∼R 0. This implies D ∼Q 0 on f−1(U) sinceD is Cartier. Therefore, there exists a positive integer m such thatf ∗f∗OX(mD) → OX(mD) is surjective. By this surjection, we havemD ∼ f∗A for a Cartier divisor A on Y which is ample over S. Thismeans that D is π-semi-ample in the usual sense.

22 2. PRELIMINARIES

2.2. Kleiman–Mori cone

In this short section, we explain the Kleiman–Mori cone and givesome interesting examples.

2.2.1 (Kleiman–Mori cone, see [Kle]). Let X be a scheme over Cand let π : X → S be a proper morphism between schemes. Let Pic(X)be the group of line bundles on X. Take a complete curve on X whichis mapped to a point by π. For L ∈ Pic(X), we define the intersectionnumber L ·C = degC f

∗L, where f : C → C is the normalization of C.By this intersection pairing, we introduce a bilinear form

· : Pic(X)× Z1(X/S)→ Z,where Z1(X/S) is the free abelian group generated by integral curveswhich are mapped to points on S by π.

Now we have the notion of numerical equivalence both in Z1(X/S)and in Pic(X), which is denoted by ≡, and we obtain a perfect pairing

N1(X/S)×N1(X/S)→ R,where

N1(X/S) = Pic(X)/ ≡⊗R and N1(X/S) = Z1(X/S)/ ≡⊗R,namely N1(X/S) and N1(X/S) are dual to each other through thisintersection pairing. It is well known that

dimRN1(X/S) = dimRN1(X/S) <∞.

We write

ρ(X/S) = dimRN1(X/S) = dimRN1(X/S).

We define the Kleiman–Mori cone NE(X/S) of π : X → S as theclosed convex cone in N1(X/S) generated by integral curves on Xwhich are mapped to points on S by π. When S = Spec C, we drop/ Spec C from the notation, e.g., we simply write N1(X) instead ofN1(X/ Spec C).

Definition 2.2.2. An element D ∈ N1(X/S) is called π-nef (orrelatively nef for π), if D ≥ 0 on NE(X/S). When S = Spec C, wesimply say that D is nef.

Kleiman’s ampleness criterion is an important result.

Theorem 2.2.3 (Kleiman’s criterion for ampleness, see [Kle]). Letπ : X → S be a projective morphism between schemes. Then L ∈Pic(X) is π-ample if and only if the numerical class of L in N1(X/S)gives a positive function on NE(X/S) \ 0.

2.2. KLEIMAN–MORI CONE 23

Remark 2.2.4. Let π : X → S be a projective morphism betweenschemes. Let D be a Cartier divisor on X. If D is π-ample in the senseof Definition 2.1.6, then D is a π-ample Cartier divisor in the usualsense by Theorem 2.2.3.

In Theorem 2.2.3, we have to assume that π : X → S is projectivesince there are complete non-projective algebraic varieties for whichKleiman’s criterion does not hold. We recall the explicit example givenin [F9] for the reader’s convenience. For the details of this example,see [F9, Section 3].

Example 2.2.5 (see [F9, Section 3]). We fix a lattice N = Z3. Wetake lattice points

v1 = (1, 0, 1), v2 = (0, 1, 1), v3 = (−1,−1, 1),

v4 = (1, 0,−1), v5 = (0, 1,−1), v6 = (−1,−1,−1).

We consider the following fan

∆ =

〈v1, v2, v4〉, 〈v2, v4, v5〉, 〈v2, v3, v5, v6〉,〈v1, v3, v4, v6〉, 〈v1, v2, v3〉, 〈v4, v5, v6〉,and their faces

.

Then the associated toric variety X = X(∆) has the following proper-ties.

(i) X is a non-projective complete toric variety with ρ(X) = 1.(ii) There exists a Cartier divisor D on X such that D is positive

on NE(X) \ 0. In particular, NE(X) is a half line.

Therefore, Kleiman’s criterion for ampleness (see Theorem 2.2.3) doesnot hold for this X. We note that X is not Q-factorial and that thereis a torus invariant curve C ' P1 on X such that C is numericallyequivalent to zero.

If X has only mild singularities, for example, X is Q-factorial, thenit is known that Theorem 2.2.3 holds even when π : X → S is proper.However, the Kleiman–Mori cone may not have enough informationswhen π is only proper.

Example 2.2.6 (see [FP]). There exists a smooth complete toricthreefold X such that NE(X) = N1(X).

The description below helps the reader understand examples in[FP].

Example 2.2.7. Let ∆ be the fan in R3 whose rays are generatedby v1 = (1, 0, 0), v2 = (0, 1, 0), v3 = (0, 0, 1), v5 = (−1, 0,−1), v6 =

24 2. PRELIMINARIES

(−2,−1, 0) and whose maximal cones are

〈v1, v2, v3〉, 〈v1, v3, v6〉, 〈v1, v2, v5〉, 〈v1, v5, v6〉, 〈v2, v3, v5〉, 〈v3, v5, v6〉.

Note that v1 + v3 + v5 = 0 and v2 + v6 = −1v1 + 1v3 + 1v5. Thus theassociated toric variety X1 = X(∆) is

PP1(OP1(−1)⊕OP1(1)⊕OP1(1)) ' PP1(OP1 ⊕OP1(2)⊕OP1(2)).

For the details, see, for example, [Ful, Exercise in Section 2.4]. Wetake a sequence of blow-ups

Yf3−→ X3

f2−→ X2f1−→ X1,

where f1 is the blow-up along the ray v4 = (0,−1,−1) = 3v1 + v5 + v6,f2 is along

v7 = (−1,−1,−1) =1

3(2v4 + v5 + v6),

and the final blow-up f3 is along the ray

v8 = (−2,−1,−1) =1

2(v5 + v6 + v7).

Then we can directly check that Y is a smooth projective toric varietywith ρ(Y ) = 5.

Finally, we remove the wall 〈v1, v5〉 and add the new wall 〈v2, v4〉.Then we obtain a flop φ : Y 99K X. We note that v2 + v4 − v1 − v5 =0. The toric variety X is nothing but X(Σ) given in [FP, Example1]. Thus, X is a smooth complete toric variety with ρ(X) = 5 andNE(X) = N1(X). Therefore, a simple flop φ : Y 99K X completelydestroys the projectivity of Y . Note that every nef line bundle on X istrivial by NE(X) = N1(X).

2.3. Singularities of pairs

We quickly review singularities of pairs in the minimal model pro-gram (see, for example, [F12], [Ko8], [Ko13], [KoMo], and so on).We also review the negativity lemmas. They are very important in theminimal model program.

First, let us recall the definition of discrepancy and total discrep-ancy of the pair (X,∆).

Definition 2.3.1 (Canonical divisor). Let X be a normal varietyof dimension n. The canonical divisor KX on X is a Weil divisor suchthat OXreg(KX) ' Ωn

Xreg, where Xreg is the smooth locus of X. Note

that the canonical divisor KX is well-defined up to linear equivalence.

2.3. SINGULARITIES OF PAIRS 25

Definition 2.3.2 (Discrepancy). Let (X,∆) be a pair where Xis a normal variety and ∆ is an R-divisor on X such that KX + ∆ isR-Cartier. Suppose f : Y → X is a resolution. We can choose Weildivisors KY and KX such that f∗KY = KX . Then, we can write

KY = f∗(KX + ∆) +∑i

a(Ei, X,∆)Ei.

This formula means that∑

i a(Ei, X,∆)Ei is defined by∑i

a(Ei, X,∆)Ei = KY − f ∗(KX + ∆).

Note that f ∗(KX +∆) is a well-defined R-Cartier R-divisor on Y sinceKX +∆ is R-Cartier. The real number a(E,X,∆) is called discrepancyof E with respect to (X,∆). The discrepancy of (X,∆) is given by

discrep(X,∆) = infEa(E,X,∆) |E is an exceptional divisor over X.

The total discrepancy of (X,∆) is given by

totaldiscrep(X,∆) = infEa(E,X,∆) |E is a divisor over X.

We note that it is indispensable to understand how to calculatediscrepancies for the study of the minimal model program.

Lemma 2.3.3 ([KoMo, Corollary 2.31 (1)]). Let X be a normalvariety and let ∆ be an R-divisor on X such that KX +∆ is R-Cartier.Then, either

discrep(X,∆) = −∞or

−1 ≤ totaldiscrep(X,∆) ≤ discrep(X,∆) ≤ 1.

Proof. Note that totaldiscrep(X,∆) ≤ discrep(X,∆) is obviousby definition. By taking a blow-up whose center is of codimensiontwo, intersects the set of smooth points of X, and is not contained inSupp ∆, we see that discrep(X,∆) ≤ 1. We assume that E is a primedivisor over X such that a(E,X,∆) = −1 − c with c > 0. We take abirational morphism f : Y → X from a smooth variety Y such that Eis a prime divisor on Y . We put

KY + ∆Y = f ∗(KX + ∆).

Let Z0 be a codimension two subvariety contained in E but not in anyother f -exceptional divisors with Z0 6⊂ Supp f−1

∗ ∆. By shrinking Y ,we may assume that E and Z0 are smooth. Let g1 : Y1 → Y be theblow-up along Z0 and let E1 be the exceptional divisor of g1. Then

a(E1, X,∆) = a(E1, Y,∆Y ) = −c.

26 2. PRELIMINARIES

Let Z1 ⊂ Y1 be the intersection of E1 and the strict transform of E.Let g2 : Y2 → Y1 be the blow-up along Z1 and let E2 ⊂ Y2 be theexceptional divisor of g2. Then

a(E2, X,∆) = a(E2, Y,∆Y ) = −2c.

By taking the blow-up whose center is the intersection of Ei and thestrict transform of E as above for i ≥ 2 repeatedly, we obtain a primedivisor Ej over X such that

a(Ej, X,∆) = −jcfor every j ≥ 1. Therefore, we obtain

discrep(X,∆) = −∞when totaldiscrep(X,∆) < −1.

Next, let us recall the basic definition of singularities of pairs.

Definition 2.3.4 (Singularities of pairs). Let (X,∆) be a pairwhere X is a normal variety and ∆ is an effective R-divisor on X suchthat KX + ∆ is R-Cartier. We say that (X,∆) is

terminal

canonical

klt

plt

lc

if discrep(X,∆)

> 0,

≥ 0,

> −1 and b∆c = 0,

> −1,

≥ −1.

Here, plt is short for purely log terminal, klt is short for kawamata logterminal, and lc is short for log canonical.

Remark 2.3.5 (Log terminal singularities). If ∆ = 0, then thenotions klt, plt, and dlt (see Definition 2.3.16 below) coincide. In thiscase, we say that X has log terminal (lt, for short) singularities.

For some inductive arguments, the notion of sub klt and sub lc isalso useful.

Definition 2.3.6 (Sub klt pairs and sub lc pairs). Let (X,∆) be apair where X is a normal variety and ∆ is a (not necessarily effective)R-divisor on X such that KX + ∆ is R-Cartier. We say that (X,∆) is

sub klt

sub lcif totaldiscrep(X,∆)

> −1

≥ −1.

Here, sub klt is short for sub kawamta log terminal and sub lc is shortfor sub log canonical.

2.3. SINGULARITIES OF PAIRS 27

It is obvious that if (X,∆) is sub lc (resp. sub klt) and ∆ is effectivethen (X,∆) is lc (resp. klt).

Remark 2.3.7. In [KoMo, Definition 2.34], ∆ is not assumed to beeffective for the definition of terminal, canonical, klt, and plt. There-fore, klt (resp. lc) in [KoMo, Definition 2.34] is nothing but sub klt(resp. sub lc) in this book.

The following lemma is well known and is very useful.

Lemma 2.3.8. Let X be a normal variety and let ∆ be an R-divisoron X such that KX + ∆ is R-Cartier. If there exists a resolutionf : Y → X such that Supp f−1

∗ ∆ ∪ Exc(f) is a simple normal crossingdivisor on Y and that

KY = f∗(KX + ∆) +∑i

a(Ei, X,∆)Ei.

If a(Ei, X,∆) > −1 for every i, then (X,∆) is sub klt. If a(Ei, X,∆) ≥−1 for every i, then (X,∆) is sub lc.

Proof. It easily follows from Lemma 2.3.9. Lemma 2.3.9 ([KoMo, Corollary 2.31 (3)]). Let X be a smooth

variety and let ∆ =∑m

i=1 ai∆i be an R-divisor such that∑

i ∆i is asimple normal crossing divisor, ai ≤ 1 for every i, and ∆i is a smoothprime divisor for every i. Then

discrep(X,∆) = min1,mini

(1− ai), mini6=j,∆i∩∆j 6=∅

(1− ai − aj)

Proof. Let r(X,∆) be the right hand side of the equality. It iseasy to see that discrep(X,∆) ≤ r(X,∆). Let E be an exceptionaldivisor for some birational morphism f : Y → X. We have to showa(E,X,∆) ≥ r(X,∆). We note that r(X,∆) does not decrease if weshrink X. Without loss of generality, we may assume that f is projec-tive, Y is smooth, and X is affine. By elimination of indeterminacy ofthe rational map f−1 : X 99K Y , we may assume that E is obtained bya succession of blow-ups along smooth irreducible centers which havesimple normal crossings with the union of the exceptional divisors andthe inverse image of Supp ∆ (see, for example, [Ko9, Corollary 3.18and Theorem 3.35]). We write t to denote the number of the blow-ups.

Xt

// Xt−1// · · · // X1

f1 // X

Y

f

22ffffffffffffffffffffffffffffffffffff

28 2. PRELIMINARIES

Let C be the center of the first blow-up f1 : X1 → X. After renumber-ing the ∆i, we may assume that codimXC = k ≥ 2 and that C ⊂ ∆i ifand only if i ≤ b for some b ≤ k. Let E1 be the exceptional divisor off1 : X1 → X. Then

a(E1, X,∆) = k − 1−∑l≤b

al.

(i) If b ≤ 0, then a(E1, X,∆) ≥ 1 ≥ r(X,∆).(ii) If b = 1, then a(E1, X,∆) ≥ 1− a1 ≥ r(X,∆).(iii) If b ≥ 2, then we have

a(E1, X,∆) ≥ (k − b− 1) +∑

1≤l≤b

(1− al)

≥ −1 + (1− a1) + (1− a2) ≥ r(X,∆).

Thus, the case where t = 1 is settled. On the other hand, if we define∆1 on X1 by

KX1 + ∆1 = f1∗(KX + ∆),

then

r(X1,∆1) ≥ minr(X,∆), 1 + a(E1, X,∆)− max∆i∩C 6=∅

ai

≥ minr(X,∆), a(E1, X,∆) ≥ r(X,∆).

Note that Supp ∆1 is a simple normal crossing divisor and the coeffi-cient of E1 in ∆1 is −a(E1, X,∆) ≤ 1. Therefore, we have

a(E,X,∆) ≥ r(X1,∆1) ≥ r(X,∆)

by induction on t. Lemma 2.3.10. Let X be a normal variety and let ∆ be an R-

divisor on X such that KX + ∆ is R-Cartier. Then there exists thelargest nonempty Zariski open set U (resp. V ) of X such that (X,∆)|Uis sub lc (resp. (X,∆)|V is sub klt).

Proof. Let f : Y → X be a resolution such that Supp f−1∗ ∆ ∪

Exc(f) is a simple normal crossing divisor on Y and that

KY = f ∗(KX + ∆) +∑i

aiEi.

We put

U = X \∪

ai<−1

f(Ei)

andV = X \

∪ai≤−1

f(Ei).

2.3. SINGULARITIES OF PAIRS 29

Then we can check that U and V are the desired Zariski open sets byLemma 2.3.9.

2.3.11 (Multiplier ideal sheaf and non-lc ideal sheaf). Let X bea normal variety and let ∆ be an effective R-divisor on X such thatKX + ∆ is R-Cartier. Let f : Y → X be a resolution with

KY + ∆Y = f∗(KX + ∆)

such that Supp ∆Y is a simple normal crossing divisor on Y .We put

J (X,∆) = f∗OY (−b∆Y c).Then J (X,∆) is an ideal sheaf on X and is known as the multiplierideal sheaf associated to the pair (X,∆). For the details, see [La2,Part Three]. It is independent of the resolution f : Y → X by theproof of Proposition 6.3.1. The closed subscheme Nklt(X,∆) definedby J (X,∆) is called the non-klt locus of (X,∆). It is obvious that(X,∆) is klt if and only if J (X,∆) = OX .

We putJNLC(X,∆) = f∗OY (−b∆Y c+ ∆=1

Y )

and call it the non-lc ideal sheaf associated to the pair (X,∆). Forthe details, see [F20]. It is independent of the resolution f : Y →X by Proposition 6.3.1. The closed subscheme Nlc(X,∆) defined byJNLC(X,∆) is called the non-lc locus of (X,∆). It is obvious that(X,∆) is log canonical if and only if JNLC(X,∆) = OX .

In the recent minimal model program, the notion of log canonicalcenters plays an important role.

Definition 2.3.12 (Log canonical centers). Let X be a normalvariety and let ∆ be an R-divisor on X such that KX +∆ is R-Cartier.Let U be the Zariski open set as in Lemma 2.3.10. If there exist aresolution f : Y → X and a divisor Ei0 on Y such that a(Ei0 , X,∆) =−1 and f(Ei0)∩U 6= ∅. Then C = f(Ei0) is called a log canonical center(an lc center, for short) of the pair (X,∆). A log canonical center whichis a minimal element with respect to the inclusion is called a minimallog canonical center (a minimal lc center, for short).

The notion of log canonical strata is useful in this book.

Definition 2.3.13 (Log canonical strata). Let X be a normal va-riety and let ∆ be an R-divisor on X such that KX +∆ is R-Cartier. Aclosed subset W of X is called a log canonical stratum (an lc stratum,for short) of the pair (X,∆) if W is X itself or is a log canonical centerof the pair (X,∆).

30 2. PRELIMINARIES

We note the notion of non-klt centers.

Definition 2.3.14 (Non-klt centers). Let X be a normal varietyand let ∆ be an R-divisor on X such that KX + ∆ is R-Cartier. Ifthere exist a resolution f : Y → X and a divisor Ei0 on Y such thata(Ei0 , X,∆) ≤ −1. Then C = f(Ei0) is called a non-klt center of thepair (X,∆).

It is obvious that any log canonical center is a non-klt center. How-ever, a non-klt center is not always a log canonical center.

2.3.15 (Divisorial log terminal pairs). Let us recall the definition ofdivisorial log terminal pairs.

Definition 2.3.16 (Divisorial log terminal pairs). Let X be a nor-mal variety and let ∆ be a boundary R-divisor such that KX + ∆ isR-Cartier. If there exists a resolution f : Y → X such that

(i) both Exc(f) and Exc(f) ∪ Supp(f−1∗ ∆) are simple normal

crossing divisors on Y , and(ii) a(E,X,∆) > −1 for every exceptional divisor E ⊂ Y ,

then (X,∆) is called divisorial log terminal (dlt, for short).

The assumption that Exc(f) is a divisor in Definition 2.3.16 (i) isvery important. See Example 3.13.9 below.

Remark 2.3.17. By Lemma 2.3.9, it is easy to see that a dlt pair(X,∆) is log canonical.

Remark 2.3.18. In Definition 2.3.16, we can require that f is pro-jective and can further require that there is an f -ample Cartier divisorA on Y whose support coincides with Exc(f). Moreover, we can makef an isomorphism over the generic point of every log canonical centerof (X,∆). For the details, see the proof of Proposition 2.3.20 below.

Lemma 2.3.19 is very useful and is indispensable for the recent min-imal model program. We sometimes call it Szabo’s resolution lemma(see [Sz] and [F12]). For more general results, see [BM] and [BVP](see also Theorem 5.2.16 and Theorem 5.2.17). Note that [Mus] is avery accessible account of the resolution of singularities.

Lemma 2.3.19 (Resolution lemma). Let X be a smooth variety andlet D be a reduced divisor on X. Then there exists a proper birationalmorphism f : Y → X with the following properties:

(1) f is a composite of blow-ups of smooth subvarieties,(2) Y is smooth,

2.3. SINGULARITIES OF PAIRS 31

(3) f−1∗ D∪Exc(f) is a simple normal crossing divisor, where f−1

∗ Dis the strict transform of D on Y , and

(4) f is an isomorphism over U , where U is the largest open setof X such that the restriction D|U is a simple normal crossingdivisor on U .

Note that f is projective and the exceptional locus Exc(f) is of purecodimension one in Y since f is a composite of blow-ups.

Proposition 2.3.20 (cf. [Sz]). Let X be a normal variety and let∆ be a boundary R-divisor on X such that KX +∆ is R-Cartier. Then(X,∆) is dlt if and only if there is a closed subset Z ⊂ X such that

(i) X \ Z is smooth and Supp ∆|X\Z is a simple normal crossingdivisor.

(ii) If h : V → X is birational and E is a prime divisor on V suchthat h(E) ⊂ Z, then a(E,X,∆) > −1.

Proof. We assume the properties (i) and (ii). By using Hiron-aka’s resolution and Szabo’s resolution lemma (see Lemma 2.3.19), wecan take a resolution f : Y → X which is a composition of blow-upsand is an isomorphism over X \ Z such that Exc(f) ∪ Supp f−1

∗ ∆ isa simple normal crossing divisor on Y by (i). By construction, f isprojective and Exc(f) is a divisor. By (ii), a(E,X,∆) > −1 for everyf -exceptional divisor E. Therefore, (X,∆) is dlt by definition. Byconstruction, it is obvious that f is an isomorphism over the genericpoint of every log canonical center of (X,∆). Note that we can take anf -ample Cartier divisor A on Y whose support coincides with Exc(f)since f is a composition of blow-ups.

Conversely, we assume that (X,∆) is dlt. Let f : Y → X be aresolution as in Definition 2.3.16. We put Z = f(Exc(f)). Then Zsatisfies the property (i). We put KY + ∆Y = f ∗(KX + ∆). Notethat f−1(Z) = Exc(f). Let ∆′ be an effective Cartier divisor whosesupport equals Exc(f). We note that every irreducible component of∆′ has coefficient < 1 in ∆Y . Therefore, (Y,∆Y + ε∆′) is sub lc for0 < ε 1 by Lemma 2.3.9. If E is any divisor over X whose centeris contained in Z, then cY (E), the center of E on Y , is contained inExc(f). Therefore, we have

a(E,X,∆) = a(E, Y,∆Y ) > a(E, Y,∆Y + ε∆′) ≥ −1.

This implies the property (ii).

The notion of weak log-terminal singularities was introduced in[KMM, Definition 0-2-10].

32 2. PRELIMINARIES

Definition 2.3.21 (Weak log-terminal singularities). Let X be anormal variety and let ∆ be a boundary R-divisor on X such thatKX + ∆ is R-Cartier. Then the pair (X,∆) is said to have weak log-terminal singularities if the following conditions hold.

(i) There exists a resolution of singularities f : Y → X such thatSupp f−1

∗ ∆∪Exc(f) is a normal crossing divisor on Y and that

KY = f ∗(KX + ∆) +∑i

aiEi

with ai > −1 for every exceptional divisor Ei.(ii) There is an f -ample Cartier divisor A on Y whose support

coincides with Exc(f).

It is easy to see that (X,∆) is log canonical when (X,∆) has weaklog-terminal singularities. We note that −A is effective by Lemma2.3.26 below.

Remark 2.3.22. By Remark 2.3.18, a dlt pair (X,∆) has weaklog-terminal singularities.

Although the notion of weak log-terminal singularities is not nec-essary for the recent developments of the minimal model program, weinclude it here for the reader’s convenience because [KMM] was writ-ten by using weak log-terminal singularities.

2.3.23 (Negativity lemmas). The negativity lemmas are very usefulin many situations. There were many papers discussing various relatedtopics before the minimal model theory appeared (see, for example,[Mum1], [Gra], [Z], and so on). Here, we closely follow the treatmentin [KoMo]. Note that Fujita’s treatment is also useful (see [Ft4, (1.5)Lemma]).

Let us start with Lemma 2.3.24, which is a special case of the Hodgeindex theorem.

Lemma 2.3.24 (see [KoMo, Lemma 3.40]). Let f : Y → X be aproper birational morphism from a smooth surface Y onto a normalsurface X with exceptional curves Ei. Assume that f(Ei) = P forevery i. Then the intersection matrix (Ei · Ej) is negative definite.

Proof. We shrink and compactify X. Then we may assume thatX and Y are projective. Let D =

∑eiEi be a non-zero linear combi-

nation of f -exceptional curves Ei. It is sufficient to prove that D2 < 0.When D is not effective, we write D = D+−D− as a difference of twoeffective divisors without common irreducible components. Then wehave

D2 ≤ D2+ +D2

−.

2.3. SINGULARITIES OF PAIRS 33

Therefore, it is sufficient to consider the case where D is effective.Assume that D2 ≥ 0. Let H be an ample Cartier divisor on Y suchthat H −KY is ample. By Serre duality, we have

H2(Y,OY (nD +H)) = 0

for every positive integer n. Note that

(nD +H · nD +H −KY ) ≥ (nD +H · nD) ≥ n(D ·H) > 0.

By the Riemann–Roch formula, we obtain that

dimH0(Y,OY (nD +H))→∞

when n→∞. On the other hand,

H0(Y,OY (nD +H)) ⊂ H0(X,OX(f∗(nD +H)))

= H0(X,OX(f∗H))

gives a contradiction. Therefore, D2 < 0. Lemma 2.3.25 (see [KoMo, Lemma 3.41]). Let Y be a smooth

surface and let C = ∪Ci be a finite set of proper curves on Y . Assumethat the intersection matrix (Ci · Cj) is negative definite. Let A =∑aiCi be an R-linear combination of the curves Ci. Assume that (A ·

Ci) ≥ 0 for every i. Then

(i) ai ≤ 0 for every i.(ii) If C is connected, then either ai = 0 for every i or ai < 0 for

every i.

Proof. We write A = A+ − A− as a difference of two effectiveR-divisors without common irreducible components. We assume thatA+ 6= 0. Since the matrix (Ci ·Cj) is negative definite, we have A2

+ < 0.Therefore, there is a curve Ci0 ⊂ SuppA+ such that (Ci0 ·A) < 0. ThenCi0 is not in SuppA−. Thus (Ci0 ·A) < 0. This is a contradiction. Weobtain (i).

We assume that C is connected, A− 6= 0, and SuppA− 6= SuppC.Then there is a curve Ci such that Ci 6⊂ SuppA− but Ci intersectsSuppA−. Then (Ci ·A) = −(Ci ·A−) < 0. This is a contradiction. Weobtain (ii).

Lemma 2.3.26 is well known as the negativity lemma.

Lemma 2.3.26 (Negativity lemma, see [KoMo, Lemma 3.39]). Letf : V → W be a proper birational morphism between normal varieties.Let −D be an f -nef R-Cartier R-divisor on V . Then

(i) D is effective if and only if f∗D is effective.

34 2. PRELIMINARIES

(ii) Assume that D is effective. Then, for every w ∈ W , eitherf−1(w) ⊂ SuppD or f−1(w) ∩ SuppD = ∅.

Proof. Note that if D is effective then so is f∗D. From now on,we assume that f∗D is effective. By Chow’s lemma and Hironaka’sresolution of singularities, there is a proper birational morphism p :V ′ → V such that V ′ → W is projective. Note that D is effective ifand only if p∗D is effective. Therefore, by replacing V with V ′, we mayassume that f is projective and V is smooth. We may further assumethat W is affine by taking an affine cover of W . We write D =

∑Dk

where Dk is the sum of those irreducible components Di of D such thatf(Di) has codimension k in W .

First, we treat the case when dimW = 2. In this case, D = D1+D2

and D1 is f -nef. Note that D1 is effective by the assumption thatf∗D is effective. Therefore, −D2 is f -nef and is a linear combinationof f -exceptional curves. By Lemma 2.3.24 and Lemma 2.3.25, D2 iseffective. This implies that D is effective when dimW = 2.

Next, we treat the general case. Let S ⊂ W be the completeintersection of dimW −2 general hypersufaces with T = f−1(S). Thenf : T → S is a birational morphism from a smooth surface T ontoa normal surface S. Note that D|T = D2|T + D1|T . Therefore, D2

is effective. Let H ⊂ V be a general very ample Cartier divisor. Weput B = D|H . Then −B is f -nef, Bi = Di+1|H for i ≥ 2 and B1 =D1|H +D2|H . Note that D1 is effective by the assumption that f∗D iseffective. We have proved that D2 is effective. Thus B1 is effective. Byinduction on the dimension, we can check that B is effective. Therefore,D is effective.

Finally, for w ∈ W , f−1(w) is connected. Thus, if f−1(w) intersectsSuppD but is not contained in it, then there is an irreducible curveC ⊂ f−1(w) such that (C · D) > 0. This is impossible since −D isf -nef. Therefore, either f−1(w) ⊂ SuppD or f−1(w) ∩ SuppD = ∅holds.

As an easy application of Lemma 2.3.26, we obtain a very usefullemma. We repeatedly use Lemma 2.3.27 and its proof in the minimalmodel theory.

Lemma 2.3.27 (see [KoMo, Lemma 3.38]). Let us consider a com-mutative diagram

X

f @@@

@@@@

@φ //_______ X ′

f ′~~

Y

2.3. SINGULARITIES OF PAIRS 35

where X, X ′, and Y are normal varieties, and f and f ′ are proper bi-rational morphisms. Let ∆ (resp. ∆′) be an R-divisor on X (resp. X ′).Assume the following conditions.

(i) f∗∆ = f ′∗∆′.

(ii) −(KX + ∆) is R-Cartier and f -nef.(iii) KX′ + ∆′ is R-Cartier and f ′-nef.

Then we have

a(E,X,∆) ≤ a(E,X ′,∆′)

for an arbitrary exceptional divisor E over Y .If either

(iv) −(KX +∆) is f -ample and f is not an isomorphism above thegeneric point of cY (E), or

(v) KX′ + ∆′ is f ′-ample and f ′ is not an isomorphism above thegeneric point of cY (E).

holds, then we have

a(E,X,∆) < a(E,X ′,∆′).

Note that cY (E) is the center of E on Y .

Proof. We take a common resolution

Zg

~~~~~~

~~~ g′

AAA

AAAA

A

Xφ

//_______ X ′

of X and X ′ such that cZ(E), the center of E on Z, is a divisor. Weput h = f g = f ′ g′. We have

KZ = g∗(KX + ∆) +∑

a(Ei, X,∆)

and

KZ = g∗(KX′ + ∆′) +∑

a(Ei, X′,∆′).

We put

H =∑

(a(Ei, X′,∆′)Ei − a(E,X,∆))Ei.

Then −H is h-nef and a sum of h-exceptional divisors by assumption(i). Therefore, H is an effective divisor by Lemma 2.3.26. Moreover,if H is not numerically h-trivial over the generic point of cY (E), thenthe coefficient of E in H is positive by Lemma 2.3.26.

We will repeatedly use the results and the arguments in this sectionthroughout this book.

36 2. PRELIMINARIES

2.4. Iitaka dimension, movable and pseudo-effective divisors

Let us start with the definition of the Iitaka dimension and thenumerical dimension. For the details, see, for example, [La1, Section2.1], [Mo4], [Nak2], [KMM, Definition 6-1-1], [U], and so on.

Definition 2.4.1 (Iitaka dimension and numerical dimension). LetX be a normal complete irreducible variety and let D be a Q-Cartierdivisor on X. Assume that m0D is Cartier for a positive integer m0.Let

Φ|mm0D| : X 99K Pdim |mm0D|

be rational mappings given by linear systems |mm0D| for positive in-tegers m. We define the Iitaka dimension or the D-dimension

κ(X,D) =

maxm>0

dim Φ|mm0D|(X), if |mm0D| 6= ∅ for some m > 0,

−∞, otherwise.

In case D is nef, we can also define the numerical dimension or thenumerical Iitaka dimension

ν(X,D) = max e |De 6≡ 0,where ≡ denotes numerical equivalence. We note that

ν(X,D) ≥ κ(X,D)

always holds. We also note that the numerical dimension ν(X,D) alsomakes sense for nef R-Cartier divisors D.

2.4.2 (Movable divisors and movable cone). We quickly review thenotion of movable divisors. It sometimes plays important roles in theminimal model program.

Definition 2.4.3 (Movable divisors and movable cone, see [Ka3,Section 2]). Let f : X → Y be a projective morphism from a normalvariety X onto a variety Y . A Cartier divisor D on X is called f -movable or movable over Y if f∗OX(D) 6= 0 and if the cokernel of thenatural homomorphism

f ∗f∗OX(D)→ OX(D)

has a support of codimension ≥ 2.Let M be an R-Cartier divisor on X. Then M is called f -movable

or movable over Y if and only if M =∑

i aiDi where ai is a positivereal number and Di is an f -movable Cartier divisor for every i.

We define Mov(X/Y ) as the closed convex cone in N1(X/Y ), whichis called the movable cone of f : X → Y , generated by the numericalclasses of f -movable Cartier divisors.

2.4. IITAKA DIMENSION, MOVABLE AND PSEUDO-EFFECTIVE DIVISORS37

Lemma 2.4.4 is a variant of the negativity lemma (see Lemma2.3.26). We will use Lemma 2.4.4 in the proof of dlt blow-ups (seeTheorem 4.4.21).

Lemma 2.4.4 (see [F26, Lemma 4.2]). Let f : X → Y be a pro-jective birational morphism from a normal Q-factorial variety X ontoa normal variety Y . Let E be an R-divisor on X such that SuppE isf -exceptional and E ∈ Mov(X/Y ). Then −E is effective.

Proof. We write E = E+−E− such that E+ and E− are effectiveR-divisors and have no common irreducible components. We assumethat E+ 6= 0. By taking a resolution of singularities, we may assumethat X is smooth. Without loss of generality, we may assume that Yis affine by taking an affine open covering of Y . Let A be an ampleCartier divisor on Y and let H be an ample Cartier divisor on X. Thenwe can find an irreducible component E0 of E+ such that

E0 · (f∗A)k ·Hn−k−2 · E < 0

when dimX = n and codimY f(E+) = k by Lemma 2.3.24. This is acontradiction. Note that

E0 · (f ∗A)k ·Hn−k−2 · E ≥ 0

since E ∈ Mov(X/Y ). Therefore, −E is effective.

2.4.5 (Pseudo-effective divisors). Let us recall the definition of pseudo-effective divisors. The notion of pseudo-effective divisors is indispens-able for the recent developments of the minimal model program, al-though it is not so important in this book.

Definition 2.4.6 (Pseudo-effective divisors). Let X be a completevariety. We define PE(X) as the closed convex cone in N1(X), whichis called the pseudo-effective cone of X, generated by the numericalclasses of effective Cartier divisors on X. Let D be an R-Cartier divisoron X. Then D is called pseudo-effective if the numerical class of D iscontained in PE(X).

Let f : X → Y be a proper morphism from a variety X to ascheme Y . Let D be an R-Cartier divisor on X. Then D is calledf -pseudo-effective or pseudo-effective over Y if the restriction of D tothe geometric generic fiber of every irreducible component of f(X) ispseudo-effective.

Although we do not need the following lemma explicitly in thisbook, it may help us understand the notion of pseudo-effective divisors.So we include it here for the reader’s convenience.

38 2. PRELIMINARIES

Lemma 2.4.7. Let f : X → Y be a projective surjective morphismbetween normal irreducible varieties with connected fibers. Let D be anR-Cartier divisor on X. Then D is pseudo-effective over Y if and onlyif D + A is big over Y for any f -ample R-Cartier divisor A on X.

Proof. First, we prove ‘if’ part. Let D be an R-Cartier divisor onX and let H be an f -ample Cartier divisor on X. Then D + 1

nH is

f -big for every positive integer n by assumption. Then the restrictionof D+ 1

nH to the geometric generic fiber of f is big. By taking n→∞,

we see that the restriction of D to the geometric generic fiber of f ispseudo-effective. Therefore, D is pseudo-effective over Y .

Next, we prove ‘only if’ part. Let A be an f -ample R-Cartier divisoron X and let D be an f -pseudo-effective R-Cartier divisor on X. Thenthe restriction of D+A to the geometric generic fiber of f is obviouslybig. Therefore, D + A is big over Y .

Nakayama’s numerical dimension for pseudo-effective divisors playscrucial roles in the recent developments of the minimal model program.So we include it for the reader’s convenience.

Definition 2.4.8 (Nakayama’s numerical dimension, see [Nak2,Chapter V.2.5. Definition]). Let D be a pseudo-effective R-Cartier di-visor on a normal projective variety X and let A be a Cartier divisor onX. If H0(X,OX(bmDc+ A)) 6= 0 for infinitely many positive integersm, then we set

σ(D;A) = max

k ∈ Z≥0

∣∣∣∣ lim supm→∞

dimH0(X,OX(bmDc+ A))

mk> 0

.

If H0(X,OX(bmDc+A)) 6= 0 only for finitely many m ∈ Z≥0, then weset σ(D;A) = −∞. We define Nakayama’s numerical dimension κσby

κσ(X,D) = maxσ(D;A) |A is a Cartier divisor on X.If D is a nef R-Cartier divisor on a normal projective variety X, thenD is pseudo-effective and

κσ(X,D) = ν(X,D).

We close this section with an easy remark.

Remark 2.4.9. Let X be a normal projective irreducible varietyand let D be a Cartier divisor on X. Then we have the followingproperties.

• If D is ample, then D is nef.• If D is semi-ample, then D is nef.• If D is ample, then D is semi-ample.

2.4. IITAKA DIMENSION, MOVABLE AND PSEUDO-EFFECTIVE DIVISORS39

• If D is nef, then D is pseudo-effective.• If D is effective, then D is pseudo-effective.• If D is movable, then D is linearly equivalent to an effective

Cartier divisor.• If D1 is a big R-divisor on X and D2 is a pseudo-effective

R-Cartier divisor on X, then D1 +D2 is big.

Let Y be a (not necessarily normal) projective irreducible variety.

• If B1 is a big R-divisor on Y and B2 is any R-Cartier divisoron Y , then B1 + εB2 is big for any 0 < ε 1.

CHAPTER 3

Classical vanishing theorems and someapplications

In this chapter, we discuss various classical vanishing theorems,for example, the Kodaira vanishing theorem, the Kawamata–Viehwegvanishing theorem, the Fujita vanishing theorem, and so on. They playcrucial roles for the study of higher-dimensional algebraic varieties. Wealso treat some applications. Although this chapter contains some newarguments and some new results, almost all results are standard orknown to the experts. Of course, our choice of topics is biased andreflects the author’s personal taste.

In Section 3.1, we give a proof of the Kodaira vanishing theorem forsmooth projective varieties based on the theory of mixed Hodge struc-tures on cohomology with compact support. It is a slightly differentfrom the usual one but suits for our framework discussed in Chapter 5.In Sections 3.2, 3.3, and 3.4, we prove the Kawamata–Viehweg vanish-ing theorem, the Viehweg vanishing theorem, and the Nadel vanishingtheorem. They are generalizations of the Kodaira vanishing theorem.In Section 3.5, we prove the Miyaoka vanishing theorem as an applica-tion of the Kawamata–Viehweg–Nadel vanishing theorem. Note thatthe Miyaoka vanishing theorem is the first vanishing theorem for theintegral part of Q-divisors. Section 3.6 is a quick review of Kollar’s in-jectivity, torsion-free, and vanishing theorems without proof. We willprove complete generalizations in Chapter 5. In Section 3.7, we treatEnoki’s injectivity theorem, which is a complex analytic counterpartof Kollar’s injectivity theorem. In Sections 3.8 and 3.9, we discussFujita’s vanishing theorem and its applications. In Section 3.10, wequickly review Tanaka’s vanishing theorems without proof. They arerelatively new and are Kodaira type vanishing theorems in positivecharacteristic. In Section 3.11, we prove Ambro’s vanishing theoremas an application of the argument in Section 3.1. In Section 3.12, wediscuss Kovacs’s characterization of rational singularities. Kovacs’sresult and its proof are very useful. In Section 3.13, we prove somebasic properties of divisorial log terminal pairs. In particular, we showthat every divisorial log terminal pair has only rational singularities

41

42 3. CLASSICAL VANISHING THEOREMS AND SOME APPLICATIONS

as an application of Kovacs’s characterization of rational singularities.Section 3.14 is devoted to the Elkik–Fujita vanishing theorem and itsapplication. We give a simplified proof of the Elkik–Fujita vanishingtheorem due to Chih-Chi Chou. In Section 3.15, we explain the methodof two spectral sequences of local cohomology groups. Section 3.16 isan introduction to our new vanishing theorems. We will discuss thedetails and more general results in Chapters 5 and 6.

3.1. Kodaira vanishing theorem

In this section, we give a proof of Kodaira’s vanishing theorem forprojective varieties based on the theory of mixed Hodge structures.

Let us start with the following easy lemma (see [Am2] and [F36]).We will prove generalizations of Lemma 3.1.1 in Chapter 5 (see Theo-rem 5.4.1 and Theorem 5.4.2).

Lemma 3.1.1. Let X be a smooth projective variety and let ∆ bea reduced simple normal crossing divisor on X. Let D be an effectiveCartier divisor on X such that SuppD ⊂ Supp ∆. Then the map

H i(X,OX(−D −∆))→ H i(X,OX(−∆)),

which is induced by the natural inclusion OX(−D) ⊂ OX , is surjectivefor every i. Equivalently, by Serre duality,

H i(X,OX(KX + ∆))→ H i(X,OX(KX + ∆ +D)),

which is induced by the natural inclusion OX ⊂ OX(D), is injective forevery i.

Proof. In this proof, we use the classical topology and Serre’sGAGA. We consider the following Hodge to de Rham type spectralsequence:

Ep,q1 = Hq(X,Ωp

X(log ∆)⊗OX(−∆))⇒ Hp+qc (X \∆,C).

It is well known that it degenerates at E1 by the theory of mixed Hodgestructures (see Remark 3.1.4 and Remark 3.1.5 below). This impliesthat the natural inclusion

ι!CX\∆ ⊂ OX(−∆),

where ι : X \∆→ X, induces the surjections

H ic(X \∆,C) = H i(X, ι!CX\∆)

αi−→ H i(X,OX(−∆))

for all i. Note that

ι!CX\∆ ⊂ OX(−D −∆) ⊂ OX(−∆).

3.1. KODAIRA VANISHING THEOREM 43

Therefore, αi factors as

αi : H i(X, ι!CX\∆)→ H i(X,OX(−D −∆))→ H i(X,OX(−∆))

for every i. This implies that

H i(X,OX(−D −∆))→ H i(X,OX(−∆))

is surjective for every i.

As an obvious application, we have:

Corollary 3.1.2. Let X be a smooth projective variety and let ∆be a reduced simple normal crossing divisor on X. Assume that there isan ample Cartier divisor D on X such that SuppD ⊂ Supp ∆. Then

H i(X,OX(−∆)) = 0

for every i < dimX, equivalently,

H i(X,OX(KX + ∆)) = 0

for every i > 0.

Proof. By Serre duality and Serre’s vanishing theorem, we have

H i(X,OX(−aD −∆)) = 0

for a sufficiently large and positive integer a and for every i < dimX.By Lemma 3.1.1, we obtain that H i(X,OX(−∆)) = 0 for every i <dimX. By Serre duality, we see that H i(X,OX(KX + ∆)) = 0 forevery i > 0.

For an application of Corollary 3.1.2, we will prove Ambro’s van-ishing theorem: Theorem 3.11.1

By using a standard covering trick, we can recover Kodaira’s van-ishing theorem for projective varieties from Lemma 3.1.1. We will treatthe Kodaira vanishing theorem for compact complex manifold in The-orem 3.7.4.

Theorem 3.1.3 (Kodaira vanishing theorem). Let X be a smoothprojective variety and let H be an ample Cartier divisor on X. Then

H i(X,OX(KX +H)) = 0

for every i > 0, equivalently, by Serre duality,

H i(X,OX(−H)) = 0

for every i < dimX.

44 3. CLASSICAL VANISHING THEOREMS AND SOME APPLICATIONS

Proof. We take a smooth divisor B ∈ |mH| for some positiveinteger m. Let f : V → X be the m-fold cyclic cover ramifying alongB. Then

f∗OV =m−1⊕k=0

OX(−kH).

Therefore, it is sufficient to prove that H i(V,OV (−f ∗H)) = 0 for everyi < dimV = dimX. This is because OX(−H) is a direct summand off∗OV (−f ∗H). Note that OV (f ∗H) has a section, which is the reducedpreimage of B, by construction. By iterating this process, we obtain atower of cyclic covers:

Vn → · · · → V0 → X.

By suitable choice of the ramification divisors, we may assume thatthe pull-back of H on Vn has no base points. Therefore, we reducedTheorem 3.1.3 to the case when the linear system |H| has no basepoints. Let ∆ ∈ |H| be a reduced smooth divisor on X. Then

H i(X,OX(−l∆))→ H i(X,OX(−∆))

is surjective for every i and every l ≥ 1 by Lemma 3.1.1. Therefore,H i(X,OX(−∆)) = 0 for i < dimX by Serre duality and Serre’s van-ishing theorem.

For the reader’s convenience, we give remarks on theE1-degenerationof the Hodge to de Rham type spectral sequence in the proof of Lemma3.1.1.

Remark 3.1.4. For the proof of Theorem 3.1.3, it is sufficient toassume that ∆ is smooth in Lemma 3.1.1. When ∆ is smooth, we caneasily construct the mixed Hodge complex of sheaves on X giving anatural mixed Hodge structure on H•c (X \∆,Z). From now on, we usethe notation and the framework in [PS, §3.3 and §3.4]. Let Hdg•(X)(resp. Hdg•(∆)) be a Hodge complex of sheaves on X (resp. ∆) givinga natural pure Hodge structure on H•(X,Z) (resp. H•(∆,Z)). Thenthe mixed cone

Hdg•(X,∆) := Cone(Hdg•(X)→ i∗Hdg•(∆))[−1],

where i : ∆→ X is the natural inclusion, gives a natural mixed Hodgestructure on H•c (X \ ∆,Z). For the details, see [PS, Example 3.24].We note that

0→ ΩpX(log ∆)⊗OX(−∆)→ Ωp

X → Ωp∆ → 0

is exact for every p. Therefore, we can easily see that

Ep,q1 = Hq(X,Ωp

X(log ∆)⊗OX(−∆))⇒ Hp+qc (X \∆,C)

3.1. KODAIRA VANISHING THEOREM 45

degenerates at E1 by the theory of mixed Hodge structures.

Remark 3.1.5. We put d = dimX. In the proof of Lemma 3.1.1,

Hq(X,ΩpX(log ∆)⊗OX(−∆))

is dual toHd−q(X,Ωd−p

X (log ∆))

by Serre duality. By Poincare duality,

Hp+qc (X \∆,C)

is dual toH2d−(p+q)(X \∆,C).

By Deligne (see [Del]), it is well known that

Ep,q1 = Hq(X,Ωp

X(log ∆))⇒ Hp+q(X \∆,C)

degenerates at E1. This implies that

dimHk(X \∆,C) =∑p+q=k

Hq(X,ΩpX(log ∆))

for every k. Therefore, by the above observation, we have

dimHkc (X \∆,C) =

∑p+q=k

Hq(X,ΩpX(log ∆)⊗OX(−∆))

for every k. Thus the Hodge to de Rham type spectral sequence

Ep,q1 = Hq(X \∆,Ωp

X(log ∆)⊗OX(−∆))⇒ Hp+qc (X \∆,C)

degenerates at E1.

Anyway, we will completely generalize Lemma 3.1.1 in Chapter 5.

Remark 3.1.6. It is well known that the Kodaira vanishing the-orem for projective varieties follows from the theory of pure Hodgestructures. For the details, see, for example, [KoMo, 2.4 The Kodairavanishing theorem].

By using a covering trick, we have a slight but very important gen-eralization of Kodaira’s vanishing theorem. Theorem 3.1.7 is usuallycalled Kawamata–Viehweg vanishing theorem.

Theorem 3.1.7 (Kawamata–Viehweg vanishing theorem). Let Xbe a smooth projective variety and let D be an ample Q-divisor on Xsuch that SuppD is a simple normal crossing divisor on X. Then

H i(X,OX(KX + dDe)) = 0

for every i > 0.

46 3. CLASSICAL VANISHING THEOREMS AND SOME APPLICATIONS

Remark 3.1.8. In this book, there are various formulations of theKawamata–Viehweg vanishing theorem in order to make it useful formany applications. See, for example, Theorem 3.1.7, Theorem 3.2.1,Theorem 3.2.8, Theorem 3.2.9, Theorem 3.3.1, Theorem 3.3.2, Theo-rem 3.3.4, Theorem 3.3.7, Theorem 4.1.1, Corollary 5.7.7, and so on.

Before we start the proof of Theorem 3.1.7, let us recall an easylemma.

Lemma 3.1.9. Let f : Y → X be a finite morphism between n-dimensional normal irreducible varieties. Then the natural inclusionOX → f∗OY is a split injection.

Proof. It is easy to see that 1nTraceY/X splits the natural inclusion

OX → f∗OY , where TraceY/X is the trace map.

Proof of Theorem 3.1.7. We put L = dDe and ∆ = L − D.We put ∆ =

∑j aj∆j such that ∆j is a reduced and smooth (possibly

disconnected) divisor on X for every j and that aj is a positive rationalnumber for every j. It is sufficient to prove H i(X,OX(−L)) = 0 forevery i < dimX by Serre duality. We use induction on the number ofdivisors ∆j. If ∆ = 0, then Theorem 3.1.7 is nothing but Kodaira’svanishing theorem: Theorem 3.1.3. We put a1 = b/m such that bis an integer and m is a positive integer. We can construct a finitesurjective morphism p1 : X1 → X such that X1 is smooth and thatp∗1∆1 ∼ mB for some Cartier divisor B on X1 (see Lemma 3.1.10). Wemay further assume that every p∗1∆j is smooth and

∑j p∗1∆j is a simple

normal crossing divisor on X1 (see Lemma 3.1.10). It is easy to see thatH i(X,OX(−L)) is a direct summand of H i(X1, p

∗1OX(−L)) by Lemma

3.1.9. By construction, p∗1∆1 is a member of |mB|. Let p2 : X2 → X1 bethe corresponding cyclic cover. Then X2 is smooth, p∗2p

∗1∆j is smooth

for every j, and∑

j p∗2p∗1∆j is a simple normal crossing divisor on X2.

Since

p2∗OX2 =m−1⊕k=0

OX1(−kB),

we obtain

H i(X2, p∗2(p∗1OX(−L)⊗OX1(bB)))

=m−1⊕k=0

H i(X1, p∗1OX(−L)⊗OX1((b− k)B)).

3.1. KODAIRA VANISHING THEOREM 47

The k = b case shows that H i(X1, p∗1OX(−L)) is a direct summand of

H i(X2, p∗2(p∗1OX(−L)⊗OX1(bB))). Note that

p∗2(p∗1L− bB) ∼ p∗2p

∗1D +

∑j>1

aip∗2p∗1∆j.

Therefore, by induction on the number of divisors ∆j, we obtain

H i(X2, p∗2(p∗1OX(−L)⊗OX1(bB))) = 0

for every i < dimX2 = dimX. Thus we obtain the desired vanishingtheorem.

The following covering trick is due to Bloch–Gieseker (see [BlGi])and is well known (see, for example, [KoMo, Proposition 2.67]).

Lemma 3.1.10. Let X be a projective variety, let D be a Cartierdivisor on X, and let m be a positive integer. Then there is a normalvariety Y , a finite surjective morphism f : Y →X, and a Cartier divisorD′ on Y such that f ∗D ∼ mD′.

Furthermore, if X is smooth and∑

j Fj is a simple normal crossingdivisor on X, then we can choose Y to be smooth such that f ∗Fj issmooth for every j and

∑j f∗Fj is a simple normal crossing divisor on

Y .

Proof. Let π : Pn → Pn be the morphism given by

(x0 : x1 : · · · , xn) 7→ (xm0 : xm1 : · · · : xmn ).

Then π∗OPn(1) ' OPn(m).Let L be a very ample Cartier divisor on X. Then there is a mor-

phism h : X → Pn such that OX(L) ' h∗OPn(1). Let Y be thenormalization of the fiber product X ×Pn Pn sitting in the diagram:

YhY //

f

Pn

π

X

h// Pn.

If D is very ample, then we put L = D. In this case,

f ∗OX(D) ' h∗Y (π∗OPn(1)) ' h∗YOPn(m).

If X is smooth, then we consider π′ : Pn → Pn which is the com-position of π with a general automorphism of the target space Pn. ByKleiman’s Bertini type theorem (see, for example, [Har4, Chapter IIITheorem 10.8]), we can make Y smooth and

∑j f∗Fj a simple normal

crossing divisor on Y .

48 3. CLASSICAL VANISHING THEOREMS AND SOME APPLICATIONS

In general, we can write D ∼ L1 − L2 where L1 and L2 are bothvery ample Cartier divisors. By using the above argument twice, weobtain f : Y → X such that f∗Li ∼ mL′i for some Cartier divisors L′ifor i = 1, 2. Thus we obtain the desired morphism f : Y → X.

3.2. Kawamata–Viehweg vanishing theorem

In this section, we generalize Theorem 3.1.7 for the latter usage.The following theorem is well known as the Kawamata–Viehweg van-ishing theorem.

Theorem 3.2.1 (see [KMM, Theorem 1-2-3]). Let X be a smoothvariety and let π : X → S be a proper surjective morphism onto avariety S. Assume that a Q-divisor D on X satisfies the followingconditions:

(i) D is π-nef and π-big, and(ii) D has support with only normal crossings.

Then Riπ∗OX(KX + dDe) = 0 for every i > 0.

Proof. We divide the proof into two steps.

Step 1. In this step, we treat a special case.

We prove the theorem under the conditions:

(1) D is π-ample, and(2) D has support with only simple normal crossings.

We may assume that S is affine since the statement is local. Then,by Lemma 3.2.3 below, we may assume that X and S are projectiveand D is ample by replacing D with D+π∗A, where A is a sufficientlyample Cartier divisor on S.

We take an ample Cartier divisor H on S and a positive integer m.Let us consider the following spectral sequence

Ep,q2 =Hp(S,Rqπ∗OX(KX + dDe+mπ∗H))

'Hp(S,Rqπ∗OX(KX + dDe)⊗OS(mH))

⇒ Hp+q(X,OX(KX + dDe+mπ∗H)).

For every sufficiently large integer m, we have Ep,q2 = 0 for p > 0 by

Serre’s vanishing theorem. Therefore, E0,q2 = Eq

∞ holds for every q.Thus, we obtain

H0(S,Rqπ∗OX(KX + dDe+mπ∗H))

= Hq(X,OX(KX + dDe+mπ∗H)) = 0

3.2. KAWAMATA–VIEHWEG VANISHING THEOREM 49

for q > 0 by Theorem 3.1.7. Since H is ample on S and m is sufficientlylarge,

Rqπ∗OX(KX + dDe+mπ∗H)

' Rqπ∗OX(KX + dDe)⊗OS(mH)

is generated by global sections. Therefore, we obtain

Riπ∗OX(KX + dDe) = 0

for every i > 0.

Step 2. In this step, we treat the general case by using the resultobtained in Step 1.

Now we prove the theorem under the conditions (i) and (ii). Wemay assume that S is affine since the statement is local. By Kodaira’slemma (see Lemma 2.1.18) and Hironaka’s resolution theorem, we canconstruct a projective birational morphism f : Y → X from anothersmooth variety Y which is projective over S and divisors Fα’s on Y suchthat Supp f ∗D∪(∪

αFα)∪Exc(f) is a simple normal crossing divisor on Y

and that f∗D−∑δαFα is πf -ample for some δα ∈ Q with 0 < δα 1

(see also [KMM, Corollary 0-3-6]). Then by applying the result provedin Step 1 to f , we obtain

0 = Rif∗OY (KY + df ∗D −∑

δαFαe) = Rif∗OY (KY + df∗De)

for every i > 0. We can also see that

f∗OY (KY + df ∗De) ' OX(KX + dDe)

by Lemma 3.2.2 below. So, we have, by the special case treated in Step1,

0 = Ri(π f)∗OY (KY + df∗D −∑

δαFαe)

= Riπ∗(f∗OY (KY + df ∗De))= Riπ∗OX(KX + dDe)

for every i > 0. Lemma 3.2.2. Let X be a smooth variety and let D be an R-divisor

on X such that SuppD is a simple normal crossing divisor on X.Let f : Y → X be a proper birational morphism from a smooth varietyY such that Supp f ∗D ∪ Exc(f) is a simple normal crossing divisoron Y . Then we have

f∗OY (KY + df ∗De) ' OX(KX + dDe).

50 3. CLASSICAL VANISHING THEOREMS AND SOME APPLICATIONS

Proof. We put ∆ = dDe − bDc. Then ∆ is a reduced simplenormal crossing divisor on X. We can write

KY + f−1∗ ∆ = f ∗(KX + ∆) +

∑Ei: f -exceptional

a(Ei, X,∆)Ei.

We have a(Ei, X,∆) ∈ Z and a(Ei, X,∆) ≥ −1 for every i. Then

KY + f−1∗ ∆ + f ∗bDc = f ∗(KX + dDe) +

∑Ei: f -exceptional

a(Ei, X,∆)Ei.

We can easily check that

multEi

(df∗De − (f−1

∗ ∆ + f∗bDc))≥ 1

for every f -exceptional divisor Ei with a(Ei, X,∆) = −1. Thus we canwrite

KY + df ∗De = f∗(KX + dDe) + F,

where F is an effective f -exceptional Cartier divisor on Y . Therefore,we have f∗OY (KY + df ∗De) ' OX(KX + dDe).

We used the following lemma in the proof of Theorem 3.2.1. We givea detailed proof for the reader’s convenience (see also Lemma 5.5.2).

Lemma 3.2.3. Let π : X → S be a projective surjective morphismfrom a smooth variety X to an affine variety S. Let D be a Q-divisoron X such that D is π-ample and SuppD is a simple normal crossingdivisor on X. Then there exist a completion π : X → S of π : X → Swhere X and S are both projective with π|X = π and a π-ample Q-divisor D on X with D|X = D such that SuppD is a simple normalcrossing divisor on X.

Proof. Let m be a sufficiently large and divisible positive integersuch that the natural surjection

π∗π∗OX(mD)→ OX(mD)

induces an embedding of X into PS(π∗OX(mD)) over S. Let π′ : X ′ →S be an arbitrary completion of π : X → S such that X ′ and S areboth projective and X ′ is smooth. We can construct such π′ : X ′ → Sby Hironaka’s resolution theorem. Let D′ be the closure of D on X ′.We consider the natural map

π′∗π′∗OX′(mD′)→ OX′(mD′).

The image of the above map can be written as

J ⊗OX′(mD′) ⊂ OX′(mD′),

where J is an ideal sheaf on X ′ such that SuppOX′/J ⊂ X ′ \X. LetX ′′ be the normalization of the blow-up of X ′ by J and f : X ′′ → X ′

3.2. KAWAMATA–VIEHWEG VANISHING THEOREM 51

the natural map. We note that f is an isomorphism over X ⊂ X ′. Wecan write f−1J · OX′′ = OX′′(−E) for some effective Cartier divisorE on X ′′. By replacing X ′ with X ′′ and mD′ with mf ∗D′ − E, wemay assume that mD′ is π-very ample over S and is π-generated overS. Therefore, we can consider the morphism ϕ : X ′ → X ′′ over Sassociated to the surjection

π′∗π′∗OX′(mD′)→ OX′(mD′)→ 0.

We note that ϕ is an isomorphsim over S by construction. By replacingX ′ with X ′′ again, we may assume that D′ is π′-ample. By using Hiron-aka’s resolution theorem, we may further assume thatX ′ is smooth. BySzabo’s resolution lemma (see Lemma 2.3.19), we can make SuppD′a simple normal crossing divisor. Thus, we obtain desired completionsπ : X → S and D.

Remark 3.2.4. In Lemma 3.2.3, we used Szabo’s resolution lemma(see Lemma 2.3.19), which was obtained after [KMM] was written.See 3.2.5 below.

3.2.5 (Kawamata–Viehweg vanishing theorem without using Szabo’sresolution lemma). Here, we explain how to prove the Kawamata–Viehweg vanishing theorem (see Theorem 3.2.1) without using Szabo’sresolution lemma (see Lemma 2.3.19). The following proof is due toNoboru Nakayama.

Proof of Theorem 3.2.1 without Szabo’s lemma. It is suf-ficient to prove Step 1 in the proof of Theorem 3.2.1. Let π : X → Sbe a projective surjective morphism from a smooth variety X to anaffine variety S and let D be a Q-divisor on X such that D is π-ampleand that SuppD is a simple normal crossing divisor on X. Then,by taking completions, we have projective varieties X and S with aprojective morphism π : X → S such that

• X is a Zariski open dense subset of X,• S is a Zariski open dense subset of S, and• π|X is the composition of π and the open immersion S → S.

X

π

// X

π

S // S

Note that π−1(S) = X since π is proper.

Claim. In the above setting, there exist a birational morphism µ :Y → X from another smooth projective variety Y and a Q-divisor Con Y such that

52 3. CLASSICAL VANISHING THEOREMS AND SOME APPLICATIONS

(i) C is relatively nef and relatively big over S,(ii) SuppC ∪ Exc(µ) is a simple normal crossing divisor on Y ,

and(iii) C|Y = µ∗D, where Y = µ−1(X) and µ = µ|Y : Y → X is the

induced birational morphism.

Here, we have an isomorphism

OX(KX + dDe) ' µ∗OY (KY + dCe).(♣)

Proof of Claim. Note that, by Lemma 3.2.2, we can check theisomorphism (♣) by (ii) and (iii). By taking a resolution of X, we mayassume that X is smooth. Then the closure D of D in X is a Q-CartierQ-divisor. Let us consider the natural homomorphism

ϕm : π∗π∗OX(mD)→ OX(mD)

for a sufficiently large positive fixed integer m such that mD is Cartier.Then ϕm is surjective on X since D is π-ample. By taking some fur-ther blow-ups, we may assume that the image of ϕm is expressed asOX(mD−E) for an effective Cartier divisor E on X with E ∩X = ∅.Thus, we have a projective variety P over S and a morphism f : X → Pover S such that

f∗H ∼ mD − Efor a Cartier divisor H on P which is relatively ample over S. Since Dis π-ample and E ∩X = ∅, the induced morphism

f |X : X = X ×S S → P := P ×S Sis finite. In particular, f is a generically finite morphism. We set

D′ := D − 1

mE.

Then D′ is π-nef and π-big, and D′|X = D. We can take a birationalmorphism µ : Y → X from another smooth projective variety Y suchthat the union of the µ-exceptional locus and Suppµ∗(D′) is a simplenormal crossing divisor on Y . We set C := µ∗D′. Then we have adesired µ : Y → X with C.

We can easily see that we can prove Theorem 3.2.1 without usingLemma 3.2.3 when S is projective (see the proof of Theorem 3.2.1).Note that we do not have to shrink S and assume that S is affine inthe proof of Theorem 3.2.1 if S is projective. From now on, we willfreely use Theorem 3.2.1 when the target space is projective.

By applying Theorem 3.2.1, we obtain

Riµ∗OY (KY + dCe) = 0

3.2. KAWAMATA–VIEHWEG VANISHING THEOREM 53

for every i > 0 since X is projective. By applying Theorem 3.2.1 again,we obtain

Ri(π µ)∗OY (KY + dCe) = 0

for every i > 0 since S is projective. Hence, by the Leray spectralsequence, we have

Riπ∗(µ∗OY (KY + dCe)) = 0

for every i > 0. By considering the restriction to S and by the isomor-phism (♣), we have

Riπ∗OX(KX + dDe) = 0

for every i > 0. It is the desired Kawamata–Viehweg vanishing theo-rem.

Remark 3.2.6. In [Nak1, Theorem 3.7], Nakayama proved a gen-eralization of the Kawamata–Viehweg vanishing theorem (see Theorem3.2.1) in the analytic category. Of course, the proof of [Nak1, Theo-rem 3.7] does not need Szabo’s resolution lemma. For several relatedresults in the analytic category, we recommend the reader to see [F31].

As a very special case of Theorem 3.2.1, we have:

Theorem 3.2.7 (Grauert–Riemenschneider vanishing theorem). Letf : X → Y be a generically finite morphism from a smooth variety X.Then Rif∗OX(KX) = 0 for every i > 0.

Proof. Note thatKX−KX is f -nef and f -big since f is genericallyfinite. Therefore, we obtain Theorem 3.2.7 as a special case of Theorem3.2.1.

For a related result, see Lemma 3.8.7, Remark 3.8.8, and Theorem3.8.9 below.

Viehweg’s formulation of the Kawamata–Viehweg vanishing theo-rem is slightly different from Theorem 3.2.1.

Theorem 3.2.8 (Viehweg). Let X be a smooth variety and let π :X → S be a proper surjective morphism onto a variety S. Assume thata Q-divisor D on X satisfies the following conditions:

(i′) D is π-nef and dDe is π-big, and(ii) D has support with only normal crossings.

Then Riπ∗OX(KX + dDe) = 0 for every i > 0.

We note that the condition (i′) in Theorem 3.2.8 is slightly weakerthan (i) in Theorem 3.2.1. We discuss a generalization of Theorem3.2.8 in Section 3.3.

54 3. CLASSICAL VANISHING THEOREMS AND SOME APPLICATIONS

Let us generalize Theorem 3.2.1 for R-divisors. We will repeatedlyuse it in the subsequent chapters.

Theorem 3.2.9 (Kawamata–Viehweg vanishing theorem for R-divisors).Let X be a smooth variety and let π : X → S be a proper surjectivemorphism onto a variety S. Assume that an R-divisor D on X satisfiesthe following conditions:

(i) D is π-nef and π-big, and(ii) D has support with only normal crossings.

Then Riπ∗OX(KX + dDe) = 0 for every i > 0.

Proof. When D is π-ample, we perturb the coefficients of D andmay assume that D is a Q-divisor. Then, by Theorem 3.2.1, we obtainRiπ∗OX(KX + dDe) = 0 for every i > 0. By using this special case,Step 2 in the proof of Theorem 3.2.1 works without any changes. So,we obtain this theorem.

As a corollary of Theorem 3.2.9, we obtain the vanishing theoremof Reid–Fukuda type. It will play important roles in the subsequentchapters. Before we state it, we prepare the following definition.

Definition 3.2.10 (Nef and log big divisors). Let f : V → W bea proper surjective morphism from a smooth variety V to a variety Wand let B be a boundary R-divisor on V such that SuppB is a simplenormal crossing divisor. We put T = bBc. Let T =

∑mi=1 Ti be the

irreducible decomposition. Let G be an R-divisor on V . We say thatG is f -nef and f -log big with respect to (V,B) if and only if G is f -nef, f -big, and G|C is f |C-big for every C, where C is an irreduciblecomponent of Ti1 ∩ · · · ∩ Tik for some i1, · · · , ik ⊂ 1, · · · ,m.

Of course, Definition 3.2.10 is compatible with Definition 5.7.2 be-low.

Theorem 3.2.11 (Vanishing theorem of Reid–Fukuda type). LetV be a smooth variety and let B be a boundary R-divisor on V suchthat SuppB is a simple normal crossing divisor. Let f : V → W be aproper morphism onto a variety W . Assume that D is a Cartier divisoron V such that D − (KV + B) is f -nef and f -log big with respect to(V,B). Then Rif∗OV (D) = 0 for every i > 0.

Proof. We use induction on the number of irreducible componentsof bBc and on the dimension of V . If bBc = 0, then Theorem 3.2.11 fol-lows from the Kawamata–Viehweg vanishing theorem: Theorem 3.2.9.Therefore, we may assume that there is an irreducible divisor S ⊂ bBc.

3.3. VIEHWEG VANISHING THEOREM 55

We consider the following short exact sequence

0→ OV (D − S)→ OV (D)→ OS(D)→ 0.

By induction, we see that Rif∗OV (D−S) = 0 and Rif∗OS(D) = 0 forevery i > 0. Thus, we have Rif∗OV (D) = 0 for every i > 0.

Note that Theorem 3.2.11 contains Norimatsu’s vanishing theorem.

Theorem 3.2.12 (Norimatsu vanishing theorem). Let X be a smoothprojective variety and let ∆ be a reduced simple normal crossing divisoron X. Let D be an ample Cartier divisor on X. Then

H i(X,OX(KX + ∆ +D)) = 0

for every i > 0.

Of course, we can obtain Theorem 3.2.12 easily as an easy conse-quence of Kodaira’s vanishing theorem (see Theorem 3.1.3) by usinginduction on the number of irreducible components of ∆ and on the di-mension of X (see the proof of Theorem 3.2.11). Note that Kawamata[Ka1] used Theorem 3.2.12 for the proof of the Kawamata–Viehwegvanishing theorem.

3.3. Viehweg vanishing theorem

Viehweg sometimes used the following formulation of the Kawamata–Viehweg vanishing theorem (see, for example, [V2, Theorem 2.28] andTheorem 3.2.8). In this book, we call it the Viehweg vanishing theorem.

Theorem 3.3.1 (Viehweg vanishing theorem). Let X be a smoothproper variety. Let L be a line bundle, let N be a positive integer, andlet D be an effective Cartier divisor on X whose support is a simplenormal crossing divisor. Assume that LN(−D) is nef and that the sheaf

L(1) = L(−bDNc)

is big. Then

H i(X,L(1) ⊗ ωX) = 0

for every i > 0.

In this section, we quickly give a proof of a slightly generalizedViehweg vanishing theorem as an application of the usual Kawamata–Viehweg vanishing theorem. For the original approach to Theorem3.3.1, see [EsVi2, (2.13) Theorem], [EsVi3, Corollary 5.12 d)], andso on. Our proof is different from the proofs given in [EsVi2] and[EsVi3].

56 3. CLASSICAL VANISHING THEOREMS AND SOME APPLICATIONS

Theorem 3.3.2. Let π : X → S be a proper surjective morphismfrom a smooth variety X, let L be an invertible sheaf on X, and letD be an effective Cartier divisor on X such that SuppD is normalcrossing. Assume that LN(−D) is π-nef for some positive integerN and that κ(Xη, (L(1))η) = m, where Xη is the generic fiber of π,(L(1))η = L(1)|Xη , and

L(1) = L(−bDNc).

Then we haveRiπ∗(L(1) ⊗ ωX) = 0

for i > dimX − dimS −m.

We note that SuppD is not necessarily simple normal crossing. Weonly assume that SuppD is normal crossing.

Remark 3.3.3. In Theorem 3.3.2, we assume that S is a point forsimplicity. We note that κ(X,L(1)) = m does not necessarily implyκ(X,L(i)) = m for 2 ≤ i ≤ N − 1, where

L(i) = L⊗i(−b iDNc).

Therefore, Viehweg’s original arguments in [V1] depending on Bogo-molov’s vanishing theorem do not seem to work in our setting.

Let us reformulate Theorem 3.2.1 for the proof of Theorem 3.3.2.

Theorem 3.3.4 (Kawamata–Viehweg vanishing theorem). Let f :Y → X be a proper surjective morphism from a smooth variety Y andlet M be a Cartier divisor on Y . Let ∆ be an effective Q-divisor onY such that Supp ∆ is normal crossing and b∆c = 0. Assume thatM − (KY + ∆) is f -nef and f -big. Then

Rif∗OY (M) = 0

for every i > 0.

Proof. We put D = M − (KY + ∆). Then D is an f -nef andf -big Q-divisor on Y such that D = d∆e −∆ and dDe = M −KY .By Theorem 3.2.1, we obtain Rif∗OY (KX + dDe) = 0 for every i > 0.Therefore, Rif∗OY (M) = 0 for every i > 0.

Remark 3.3.5. It is obvious that Theorem 3.3.4 is a special case ofTheorem 3.3.2. By applying Theorem 3.3.2, the assumption in Theo-rem 3.3.4 can be weaken as follows: M−(KX+∆) is f -nef and M−KX

is f -big. We note that M − KX is f -big if M − (KX + ∆) is f -big.In this section, we give a quick proof of Theorem 3.3.2 only by using

3.3. VIEHWEG VANISHING THEOREM 57

Theorem 3.3.4 and Hironaka’s resolution. Therefore, Theorem 3.3.2 isessentially the same as Theorem 3.3.4.

Let us start the proof of Theorem 3.3.2.

Proof of Theorem 3.3.2. Without loss of generality, we mayassume that S is affine. Let f : Y → X be a proper birational mor-phism from a smooth quasi-projective variety Y such that Supp f∗D∪Exc(f) is a simple normal crossing divisor. We write

KY = f ∗(KX + (1− ε)DN) + Eε.

Then F = dEεe is an effective exceptional Cartier divisor on Y andindependent of ε for 0 < ε 1. Therefore, the coefficients of F − Eεare continuous for 0 < ε 1. Let L be a Cartier divisor on X suchthat L ' OX(L). We may assume that κ(Xη, (L − bDN c)η) = m ≥ 0.Let Φ : X 99K Z be the relative Iitaka fibration over S with respect tol(L−bD

Nc), where l is a sufficiently large and divisible positive integer.

We may further assume that

f∗(L− bDNc) ∼Q ϕ

∗A+ E,

where E is an effective Q-divisor such that SuppE∪Supp f ∗D∪Exc(f)is simple normal crossing, ϕ = Φ f : Y → Z is a morphism, and A isa ψ-ample Q-divisor on Z with ψ : Z → S.

Y

f

ϕ

@@@

@@@@

XΦ //___

π @

@@@@

@@@ Z

ψ

S

Let ∑i

Ei = SuppE ∪ Supp f∗D ∪ Exc(f)

be the irreducible decomposition. We can write Eε =∑

i aεiEi and

E =∑

i biEi. We note that aεi is continuous for 0 < ε 1. We put

∆ε = F − Eε + εE.

By definition, we can see that every coefficient of ∆ε is in [0, 2) for0 < ε 1. Thus, b∆εc is reduced. If aεi < 0, then aεi ≥ −1 + 1

Nfor

0 < ε 1. Therefore, if daεie−aεi +εbi ≥ 1 for 0 < ε 1, then aεi > 0.

58 3. CLASSICAL VANISHING THEOREMS AND SOME APPLICATIONS

Thus, F ′ = F − b∆εc is effective and f -exceptional for 0 < ε 1. Onthe other hand, (Y, ∆ε) is obviously klt for 0 < ε 1. We note that

f∗(KX + L− bDNc) + F ′ − (KY + ∆ε)

= f ∗(KX + L− bDNc) + F − f∗(KX + (1− ε)D

N)− Eε

− (F − Eε + εE)

∼Q (1− ε)f ∗(L− D

N) + εϕ∗A

for a rational number ε with 0 < ε 1. We put

M = f ∗(KX + L− bDNc) + F ′.

Let H be a p-ample general smooth Cartier divisor on Y , where p =ψ ϕ = π f : Y → S. Since

(M +H)− (KY + ∆ε) ∼Q (1− ε)f∗(L− D

N) + εϕ∗A+H,

is p-ample, we obtain

Rip∗OY (M +H) = 0

for every i > 0 by Theorem 3.3.4. By the long exact sequence

· · · → Rip∗OY (M)→ Rip∗OY (M +H)→ Rip∗OH(M +H)→ · · ·obtained from

0→ OY (M)→ OY (M +H)→ OH(M +H)→ 0,

we obtainRip∗OH(M +H) ' Ri+1p∗OY (M)

for every i > 0. We note that

M − (KY + ∆ε) ∼Q (1− ε)f∗(L− D

N) + εϕ∗A

and

(M +H)|H − (KH + ∆ε|H) ∼Q (1− ε)f ∗(L− D

N)|H + εϕ∗A|H .

We also note that (H, ∆ε|H) is klt and

κ(Hη, (ϕ∗A)|Hη) ≥ minm, dimHη.

By repeating the above argument, that is, taking a general smoothhyperplane cut, and by Theorem 3.3.4, we obtain

Rip∗OY (M) = Rip∗OY (f∗(KX + L− bDNc) + F ′) = 0

3.3. VIEHWEG VANISHING THEOREM 59

for every i > dimY − dimS −m = dimX − dimS −m (see also [V1,Remark 0.2]). On the other hand,

Rif∗OY (M) = Rif∗OY (f ∗(KX + L− bDNc) + F ′) = 0

for every i > 0 by Theorem 3.3.4. We note that

f∗OY (f ∗(KX + L− bDNc) + F ′) ' OX(KX + L− bD

Nc)

by the projection formula because F ′ is effective and f -exceptional.Therefore, we obtain

Riπ∗OX(KX + L− bDNc) = Rip∗OY (M) = 0

for every i > dimX − dimS −m.

We give an obvious corollary of Theorem 3.3.2.

Corollary 3.3.6. Let X be an n-dimensional smooth completevariety and let L be an invertible sheaf on X. Assume that D ∈ |LN | forsome positive integer N and that SuppD is a simple normal crossingdivisor on X. Then we have

H i(X,L(1) ⊗ ωX) = 0

for i > n− κ(X, DN).

We think that Theorem 3.3.7, which is similar to Theorem 3.3.2and easily follows from the usual Kawamata–Viehweg vanishing the-orem: Theorem 3.2.1 (see also Theorem 3.3.4), is easier to use thanTheorem 3.3.2. So we contain it for the reader’s convenience.

Theorem 3.3.7 (Kawamata–Viehweg vanishing theorem). Let f :Y → X be a projective morphism from a smooth variety Y onto avariety X. Let ∆ be an effective Q-divisor on Y such that Supp ∆ is anormal crossing divisor and that b∆c = 0. Let M be a Cartier divisoron Y such that M−(KY +∆) is f -nef and ν(Xη, (M−(KY +∆))|Xη) =m, where Xη is the generic fiber of f . Then Rif∗OY (M) = 0 for everyi > dimY − dimX −m.

Proof. We use induction on dimY −dimX. If dimY −dimX = 0,then M− (KY +∆) is f -big. Therefore, Theorem 3.3.7 is a special caseof Theorem 3.3.4 when dimY −dimX = 0. When m = dimY −dimX,Theorem 3.3.7 follows from Theorem 3.3.4. Thus, we may assume thatm < dimY − dimX. Without loss of generality, we may assume thatX is affine by shrinking X. Let A be an f -very ample Cartier divisor

60 3. CLASSICAL VANISHING THEOREMS AND SOME APPLICATIONS

on Y . We take a general member H of |A|. We consider the followingshort exact sequence

0→ OY (M)→ OY (M +H)→ OH(M +H)→ 0.

Since M +H − (KY + ∆) is f -ample, we obtain Rif∗OY (M +H) = 0for every i > 0 by Theorem 3.3.4. This implies that

Rif∗OH(M +H) ' Ri+1f∗OY (M)

holds for every i ≥ 1. Since (M +H)|H − (KH + ∆|H) is f -nef and

ν(Hη, ((M +H)|H − (KH + ∆|H))|Hη) ≥ m,

where Hη is the generic fiber of H → f(H), we obtain

Rif∗OH(M +H) = 0

for i > dimH − dimX −m = dimY − dimX −m− 1 by induction ondimY − dimX. Therefore, we have

Rif∗OY (M) = 0

for i > dimY − dimX −m.

3.4. Nadel vanishing theorem

Let us recall the definition of the multiplier ideal sheaf of a pair(X,∆) (see also 2.3.11).

Definition 3.4.1 (Multiplier ideal sheaves). Let X be a normalvariety and let ∆ be a (not necessarily effective) R-divisor on X suchthat KX + ∆ is R-Cartier. Let f : Y → X be a resolution such that

KY + ∆Y = f∗(KX + ∆)

and that Supp ∆Y is a simple normal crossing divisor on Y . We put

J (X,∆) = f∗OY (−b∆Y c)

and call it the multiplier ideal sheaf of the pair (X,∆). It is easy to seethat J (X,∆) is independent of the resolution f : Y → X by the proofof Proposition 6.3.1. When ∆ is effective, we have J (X,∆) ⊂ OX .

The following (algebraic version of) Nadel vanishing theorem is veryimportant for the recent developments of the higher-dimensional alge-braic geometry (see, for example, [HaKo, Chapter 6, Multiplier idealsheaves]). It is a variant of the Kawamata–Viehweg vanishing theorem(see, for example, Theorem 3.2.9).

3.5. MIYAOKA VANISHING THEOREM 61

Theorem 3.4.2 (Nadel vanishing theorem). Let X be a normalvariety and let ∆ be an R-divisor on X such that KX +∆ is R-Cartier.Let D be a Cartier divisor on X such that D− (KX + ∆) is π-nef andπ-big, where π : X → S is a proper surjective morphism onto a varietyS. Then

Riπ∗(OX(D)⊗ J (X,∆)) = 0

for every i > 0.

Proof. Let f : Y → X be a resolution as in Definition 3.4.1. Then

f ∗D − b∆Y c − (KY + ∆Y ) = f ∗(D − (KX + ∆))

is π f -nef and π f -big. In particular, it is f -nef and f -big. By theKawamata–Viehweg vanishing theorem (see Theorem 3.2.9), we have

Rif∗OY (f∗D − b∆Y c) = 0

for every i > 0. By the projection formula, we obtain

f∗OY (f ∗D − b∆Y c) = OX(D)⊗ J (X,∆).

By the Kawamata–Viehweg vanishing theorem (see Theorem 3.2.9)again, we have

Ri(π f)∗OY (f∗D − b∆Y c) = 0

for every i > 0. Therefore, Riπ∗(OX(D) ⊗ J (X,∆)) = 0 for everyi > 0.

Remark 3.4.3. Let X be a smooth variety and let ∆ be an effec-tive R-divisor on X such that Supp ∆ is a normal crossing divisor andthat b∆c = 0. Then we can easily check that J (X,∆) = OX . There-fore, Theorem 3.4.2 contains the usual Kawamata–Viehweg vanishingtheorem (see, for example, Theorem 3.3.4).

For the details of the theory of (algebraic) multiplier ideal sheaves,we recommend the reader to see [La2, Part Three].

3.5. Miyaoka vanishing theorem

Let us recall Miyaoka’s vanishing theorem (see [Mi, Proposition2.3]). Miyaoka’s vanishing theorem is the first vanishing theorem forthe integral part of Q-divisors. So, it is a historically important result.

Theorem 3.5.1 (Miyaoka vanishing theorem). Let X be a smoothprojective surface and let D be a big Cartier divisor on X. Let D =P + N be its Zariski decomposition, where P (resp. N) is the positive(resp. negative) part of the Zariski decomposition. Assume that bNc =0. Then H1(X,OX(−D)) = 0.

62 3. CLASSICAL VANISHING THEOREMS AND SOME APPLICATIONS

Let us quickly recall the Zariski decomposition on smooth projectivesurfaces (see [Z]). For the details, see, for example, [Ba, Chapter 14].

Theorem 3.5.2 (Zariski decomposition). Let D be a big Cartierdivisor on a smooth projective surface X. Then there exists a uniquedecomposition

D = P +N

which satisfies the following conditions:

(i) P is nef and big;(ii) N is an effective Q-divisor;(iii) P · C = 0 for every irreducible component C of SuppN .

This decomposition is called the Zariski decomposition of D. We usu-ally call P (resp. N) the positive (resp. negative) part of the Zariskidecomposition of D.

The following statement is a correct formulation of Miyaoka’s van-ishing theorem (see Theorem 3.5.1) from our modern viewpoint.

Theorem 3.5.3. Let X be a smooth complete variety with dimX ≥2 and let D be a Cartier divisor on X. Assume that D is numer-ically equivalent to M + B, where M is a nef Q-divisor on X withν(X,M) ≥ 2 and B is an effective Q-divisor with bBc = 0. ThenH1(X,OX(−D)) = 0.

Proof. By Serre duality, it is sufficient to see that

Hn−1(X,OX(KX +D)) = 0,

where n = dimX. Let J (X,B) be the multiplier ideal sheaf of (X,B)(see Definition 3.4.1). We consider

· · · → Hn−1(X,OX(KX +D)⊗ J (X,B))→ Hn−1(X,OX(KX +D))

→ Hn−1(X,OX(KX +D)⊗OX/J (X,B))→ · · · .Since bBc = 0, we see that dim SuppOX/J (X,B) ≤ n− 2. Therefore,

Hn−1(X,OX(KX +D)⊗OX/J (X,B)) = 0.

Thus, it is enough to see that

Hn−1(X,OX(KX +D)⊗ J (X,B)) = 0.

Let f : Y → X be a resolution such that Supp f∗B is a simple normalcrossing divisor. Then we have

J (X,B) = f∗OY (KY/X − bf ∗Bc)and

Rif∗OY (KY/X − bf ∗Bc) = 0

3.6. KOLLAR INJECTIVITY THEOREM 63

for every i > 0 (see, for example, Theorem 3.2.1). So, we obtain

Hn−1(X,OX(KX +D)⊗ J (X,B))

' Hn−1(Y,OY (KY + f ∗D − bf ∗Bc)) = 0

by Theorem 3.3.7. Remark 3.5.4. In Theorem 3.5.3, we can replace the assumption

ν(X,M) ≥ 2 with κ(Y, f ∗D − bf ∗Bc) ≥ 2 by Theorem 3.3.2.

Proof of Theorem 3.5.1. Since D = P + N is the Zariski de-composition, P is a nef and big Q-divisor on X and N is an effectiveQ-divisor on X. By assumption, bNc = 0. Thus, by Theorem 3.5.3,we obtain Theorem 3.5.1.

3.6. Kollar injectivity theorem

In this section, we quickly review Kollar’s injectivity theorem, torsion-free theorem, and vanishing theorem without proof.

In [Tank], Tankeev proved:

Theorem 3.6.1 ([Tank, Proposition 1]). Let X be a smooth pro-jective variety with dimX ≥ 2. Assume that the complete linear system|H| has no base points and determines a morphism Φ|H| : X → Y ontoa variety Y with dimY ≥ 2. Then

H0(X,OX(KX + 2D))→ H0(D,OD((KX + 2D)|D))

is surjective for almost all divisors D ∈ |H|.

Proof. By Bertini, D is smooth. Therefore, by Lemma 3.1.1, weobtain that

H1(X,OX(KX +D))→ H1(X,OX(KX + 2D))

is injective. Thus we obtain the desired surjection. In [Ko2], Kollar obtained Theorem 3.6.2 as a generalization of The-

orem 3.6.1. We call it Kollar’s injectivity theorem.

Theorem 3.6.2 ([Ko2, Theorem 2.2]). Let X be a smooth projec-tive variety and let L be a semi-ample Cartier divisor on X. Let D bea member of |kL|. Then

H i(X,OX(KX + nL))→ H i(X,OX(KX + (n+ k)L)),

which is induced by the natural inclusion OX ⊂ OX(D) ' OX(kL), isinjective for every i and every positive integer n.

He also obtained Theorem 3.6.3 in [Ko2].

64 3. CLASSICAL VANISHING THEOREMS AND SOME APPLICATIONS

Theorem 3.6.3 ([Ko2, Theorem 2.1]). Let X be a smooth projec-tive variety, let Y be an arbitrary projective variety, and let π : X → Ybe a surjective morphism. Then we have the following properties.

(i) Riπ∗OX(KX) is torsion-free for every i.(ii) Let L be an ample Cartier divisor on Y , then

Hj(Y,OY (L)⊗Riπ∗OX(KX)) = 0

for every j > 0 and every i.

We usually call Theorem 3.6.3 (i) (resp. (ii)) Kollar’s torsion-freetheorem (resp. Kollar’s vanishing theorem). Note that Theorem 3.6.3(ii) contains the Kodaira vanishing theorem for projective varieties: The-orem 3.1.3. We also note that Theorem 3.6.3 (i) generalizes the Grauert–Riemenschneider vanishing theorem: Theorem 3.2.7.

In [Ko2], Kollar proved Theorem 3.6.2 and Theorem 3.6.3 simulta-neously. Therefore, the relationship between Theorem 3.6.2 and Theo-rem 3.6.3 is not clear by the proof in [Ko2]. Now it is well known thatTheorem 3.6.2 and Theorem 3.6.3 are equivalent by the works of Kollarhimself and Esnault–Viehweg (see, for example, [EsVi3] and [Ko5]).For the proof of Theorem 3.6.2 and Theorem 3.6.3, see also [EsVi1].

We do not prove Kollar’s theorems here. We will prove completegeneralizations in Chapter 5.

3.7. Enoki injectivity theorem

In this section, we discuss Enoki’s injectivity theorem (see [Eno,Theorem 0.2]), which contains Kollar’s original injectivity theorem: The-orem 3.6.2. We recommend the reader to compare the proof of Theorem3.7.1 with the arguments in [Ko2, Section 2] and [Ko6, Chapter 9].

Theorem 3.7.1 (Enoki’s injectivity theorem). Let X be a compactKahler manifold and let L be a semi-positive line bundle on X. Then,for any non-zero holomorphic section s of L⊗k with some positive in-teger k, the multiplication homomorphism

×s : Hq(X,ωX ⊗ L⊗l) −→ Hq(X,ωX ⊗ L⊗(l+k)),

which is induced by ⊗s, is injective for every q ≥ 0 and l > 0.

Let us recall the basic notion of the complex geometry. For details,see, for example, [Dem].

Definition 3.7.2 (Chern connection and its curvature form). LetX be a complex manifold and let (E, h) be a holomorphic hermitianvector bundle on X. Then there exists the Chern connection D =D(E,h), which can be split in a unique way as a sum of a (1, 0) and of a

3.7. ENOKI INJECTIVITY THEOREM 65

(0, 1)-connection, D = D′(E,h) +D′′(E,h). By the definition of the Chern

connection, D′′ = D′′(E,h) = ∂. We obtain the curvature form

Θh(E) := D2(E,h).

The subscripts might be suppressed if there is no risk of confusion.Let L be a holomorphic line bundle on X. We say that L is positive

(reps. semi-positive) if there exists a smooth hermitian metric hL on Lsuch that

√−1ΘhL

(L) is a positive (resp. semi-positive) (1, 1)-form onX.

Definition 3.7.3 (Inner product). Let X be an n-dimensionalcomplex manifold with the hermitian metric g. We denote by ω thefundamental form of g. Let (E, h) be a holomorphic hermitian vec-tor bundle on X, and u, v are E-valued (p, q)-forms with measurablecoefficients, we set

‖u‖2 =

∫X

|u|2dVω, 〈〈u, v〉〉 =

∫X

〈u, v〉dVω,

where |u| (resp. 〈u, v〉) is the pointwise norm (resp. inner product)induced by g and h on Λp,qT ∗X ⊗ E, and dVω = 1

n!ωn.

Let us prove Theorem 3.7.1.

Proof of Theorem 3.7.1. Throughout this proof, we fix a Kahlermetric g on X. Let h be a smooth hermitian metric on L such that thecurvature

√−1Θh(L) =

√−1∂∂ log h is a smooth semi-positive (1, 1)-

form on X. We put n = dimX. We introduce the space of L⊗l-valuedharmonic (n, q)-forms as follows,

Hn,q(X,L⊗l) := u ∈ Cn,q(X,L⊗l)|∆′′u = 0for every q ≥ 0, where

∆′′ := ∆′′(L⊗l,hl) := D′′∗(L⊗l,hl)∂ + ∂D′′∗(L⊗l,hl)

and Cn,q(X,L⊗l) is the space of L⊗l-valued smooth (n, q)-forms on X.We note that D′′

(L⊗l,hl)= ∂ and that D′′∗

(L⊗l,hl)is the formal adjoint of

D′′(L⊗l,hl)

. It is easy to see that ∆′′u = 0 if and only if

D′′∗(L⊗l,hl)u = ∂u = 0

for u ∈ Cn,q(X,L⊗l) since X is compact. It is well known that

Cn,q(X,L⊗l) = Im∂ ⊕Hn,q(X,L⊗l)⊕ ImD′′∗(L⊗l,hl)

and

Ker∂ = Im∂ ⊕Hn,q(X,L⊗l).

66 3. CLASSICAL VANISHING THEOREMS AND SOME APPLICATIONS

Therefore, we have the following isomorphisms,

Hq(X,ωX ⊗ L⊗l) ' Hn,q(X,L⊗l) =Ker∂

Im∂' Hn,q(X,L⊗l).

We obtain Hq(X,ωX ⊗ L⊗(l+k)) ' Hn,q(X,L⊗(l+k)) similarly.

Claim. The multiplication map

×s : Hn,q(X,L⊗l) −→ Hn,q(X,L⊗(l+k))

is well-defined.

If the claim is true, then the theorem is obvious. This is becausesu = 0 in Hn,q(X,L⊗(l+k)) implies u = 0 for u ∈ Hn,q(X,L⊗l). Thisimplies the desired injectivity. Thus, it is sufficient to prove the aboveclaim.

Proof of Claim. By the Nakano identity (see, for example, [Dem,(4.6)]), we have

‖D′′∗(L⊗l,hl)u‖2 + ‖D′′u‖2 = ‖D′∗u‖2 + 〈〈

√−1Θhl(L⊗l)Λu, u〉〉

holds for L⊗l-valued smooth (n, q)-form u, where Λ is the adjoint ofω∧ · and ω is the fundamental form of g. If u ∈ Hn,q(X,L⊗l), then theleft hand side is zero by the definition of Hn,q(X,L⊗l). Thus we obtain‖D′∗u‖2 = 〈〈

√−1Θhl(L⊗l)Λu, u〉〉 = 0 since

√−1Θhl(L⊗l) =

√−1lΘh(L)

is a smooth semi-positive (1, 1)-form on X. Therefore, D′∗u = 0 and〈√−1Θhl(L⊗l)Λu, u〉hl = 0, where 〈 , 〉hl is the pointwise inner product

with respect to hl and g. By Nakano’s identity again,

‖D′′∗(L⊗(l+k),hl+k)(su)‖2 + ‖D′′(su)‖2

= ‖D′∗(su)‖2 + 〈〈√−1Θhl+k(L⊗(l+k))Λsu, su〉〉

Note that we assumed u ∈ Hn,q(X,L⊗l). Since s is holomorphic,D′′(su) = ∂(su) = 0 by the Leibnitz rule. We know that

D′∗(su) = − ∗ ∂ ∗ (su) = sD′∗u = 0

since s is a holomorphic L⊗k-valued (0, 0)-form andD′∗u = 0, where ∗ isthe Hodge star operator with respect to g. Note thatD′∗ is independentof the fiber metrics. So, we have

‖D′′∗(L⊗(l+k),hl+k)(su)‖2 = 〈〈

√−1Θhl+k(L⊗(l+k))Λsu, su〉〉.

3.7. ENOKI INJECTIVITY THEOREM 67

We note that

〈√−1Θhl+k(L⊗(l+k))Λsu, su〉hl+k

=l + k

l|s|2hk〈

√−1Θhl(L⊗l)Λu, u〉hl = 0

where 〈 , 〉hl+k (resp. |s|hk) is the pointwise inner product (resp. thepointwise norm of s) with respect to hl+k and g (resp. with respect tohk). Thus, we obtain D′′∗

(L⊗(l+k),hl+k)(su) = 0. Therefore, we know that

∆′′(L⊗(l+k),hl+k)

(su) = 0, equivalently, su ∈ Hn,q(X,L⊗(l+k)). We finish

the proof of the claim.

Thus we obtain the desired injectivity theorem.

The above proof of Theorem 3.7.1, which is due to Enoki, is ar-guably simpler than Kollar’s original proof of his injectivity theorem(see Theorem 3.6.2) in [Ko2].

We include Kodaira’s vanishing theorem for compact complex man-ifolds and its proof based on Bochner’s technique for the reader’s con-venience.

Theorem 3.7.4 (Kodaira vanishing theorem for complex mani-folds). Let X be a compact complex manifold and let L be a positiveline bundle on X. Then Hq(X,ωX ⊗ L) = 0 for every q > 0.

Proof. We take a smooth hermitian metric h on L such that√−1Θh(L) =

√−1∂∂ log h is a smooth positive (1, 1)-form on X. We

define a Kahler metric g on X associated to ω :=√−1Θh(L). As we

saw in the proof of Theorem 3.7.1, we have

Hq(X,ωX ⊗ L) ' Hn,q(X,L)

where n = dimX and Hn,q(X,L) is the space of L-valued harmonic(n, q)-forms on X. We take u ∈ Hn,q(X,L). By Nakano’s identity, wehave

0 = ‖D′′∗(L,h)u‖2 + ‖D′′u‖2

= ‖D′∗u‖2 + 〈〈√−1Θh(L)Λu, u〉〉.

On the other hand, we have

〈√−1Θh(L)Λu, u〉h = q|u|2h.

Therefore, we obtain 0 = ‖u‖2 when q ≥ 1. Thus, we have u = 0.This means that Hn,q(X,L) = 0 for every q ≥ 1. Therefore, we haveHq(X,ωX ⊗ L) = 0 for every q ≥ 1.

68 3. CLASSICAL VANISHING THEOREMS AND SOME APPLICATIONS

It is a routine work to prove Theorem 3.7.5 by using Theorem 3.7.1.More precisely, Theorem 3.6.2 for compact Kahler manifolds, which isa special case of Theorem 3.7.1, induces Theorem 3.7.5 by the usualargument as in [EsVi3] and [Ko6].

Theorem 3.7.5 (Torsion-freeness and vanishing theorem). Let Xbe a compact Kahler manifold and let Y be a projective variety. Letπ : X → Y be a surjective morphism. Then we obtain the followingproperties.

(i) Riπ∗ωX is torsion-free for every i ≥ 0.(ii) If H is an ample line bundle on Y , then

Hj(Y,H ⊗Riπ∗ωX) = 0

for every i ≥ 0 and j > 0.

For related topics, see [Take2], [Oh], [F30], and [F31]. See also[F37]. We close this section with a conjecture.

Conjecture 3.7.6. Let X be a compact Kahler manifold (or asmooth projective variety) and let D be a reduced simple normal cross-ing divisor on X. Let L be a semi-positive line bundle on X and let sbe a non-zero holomorphic section of L⊗k on X for some positive in-teger k. Assume that (s = 0) contains no strata of D, that is, (s = 0)contains no log canonical centers of (X,D). Then the multiplicationhomomorphism

×s : Hq(X,ωX ⊗OX(D)⊗ L⊗l)→ Hq(X,ωX ⊗OX(D)⊗ L⊗(l+k)),

which is induced by ⊗s, is injective for every q ≥ 0 and l > 0.

3.8. Fujita vanishing theorem

The following theorem was obtained by Takao Fujita (see [Ft1,Theorem (1)] and [Ft2, (5.1) Theorem]). See also [La1, Theorem1.4.35].

Theorem 3.8.1 (Fujita vanishing theorem). Let X be a projectivescheme defined over a field k and let H be an ample Cartier divisor onX. Given any coherent sheaf F on X, there exists an integer m(F , H)such that

H i(X,F ⊗OX(mH +D)) = 0

for all i > 0, m ≥ m(F , H), and any nef Cartier divisor D on X.

Proof. Without loss of generality, we may assume that k is alge-braically closed. By replacing X with SuppF , we may assume thatX = SuppF .

3.8. FUJITA VANISHING THEOREM 69

Remark 3.8.2. Let F be a coherent sheaf on X. In the proof ofTheorem 3.8.1, we always define a subscheme structure on SuppF bythe OX-ideal Ker(OX → EndOX

(F)).

We use induction on the dimension.

Step 1. When dimX = 0, Theorem 3.8.1 obviously holds.

From now on, we assume that Theorem 3.8.1 holds in the lowerdimensional case.

Step 2. We can reduce the proof to the case where X is reduced.

Proof of Step 2. We assume that Theorem 3.8.1 holds for re-duced schemes. Let N be the nilradical of OX , so that N r = 0 forsome r > 0. Consider the filtration

F ⊃ N · F ⊃ N 2 · F ⊃ · · · ⊃ N r · F = 0.

The quotients N iF/N i+1F are coherent OXred-modules, and therefore,

by assumption,

Hj(X, (N iF/N i+1F)⊗OX(mH +D)) = 0

for j > 0 and m ≥ m(N iF/N i+1F , H) thanks to the amplitude ofOXred

(H). Twisting the exact sequences

0→ N i+1F → N iF → N iF/N i+1F → 0

by OX(mH + D) and taking cohomology, we then find by decreasinginduction on i that

Hj(X,N iF ⊗OX(mH +D)) = 0

for j > 0 and m ≥ m(N iF , H). When i = 0 this gives the desiredvanishings.

From now on, we assume that X is reduced.

Step 3. We can reduce the proof to the case where X is irreducible.

Proof of Step 3. We assume that Theorem 3.8.1 holds for re-duced and irreducible schemes. Let X = X1 ∪ · · · ∪Xk be its decom-position into irreducible components and let I be the ideal sheaf of X1

in X. We consider the exact sequence

0→ I · F → F → F/I · F → 0.

The outer terms of the above exact sequence are supported onX2∪· · ·∪Xk and X1 respectively. So by induction on the number of irreduciblecomponents, we may assume that

Hj(X, IF ⊗OX(mH +D)) = 0

70 3. CLASSICAL VANISHING THEOREMS AND SOME APPLICATIONS

for j > 0 and m ≥ m(IF , H|X2∪···∪Hk) and

Hj(X, (F/IF)⊗OX(mH +D)) = 0

for j > 0 and m ≥ m(F/IF , H|X1). It then follows from the aboveexact sequence that

Hj(X,F ⊗OX(mH +D)) = 0

when j > 0 and

m ≥ m(F , H) := maxm(IF , H|X2∪···∪Hk),m(F/IF , H|X1),

as required. From now on, we assume that X is reduced and irreducible.

Step 4. We can reduce the proof to the case whereH is very ample.

Proof of Step 4. Let l be a positive integer such that lH is veryample. We assume that Theorem 3.8.1 holds for lH. Apply Theorem3.8.1 to F ⊗ OX(nH) for 0 ≤ n ≤ l − 1 with lH. Then we obtainm(F ⊗OX(nH), lH) for 0 ≤ n ≤ l − 1. We put

m(F , H) = l(maxn

m(F ⊗OX(nH), lH) + 1).

Then we can easily check that m(F , H) satisfies the desired property.

From now on, we assume that H is very ample.

Step 5. It is sufficient to find m(F , H) such that

H1(X,F ⊗OX(mH +D)) = 0

for all m ≥ m(F , H) and any nef Cartier divisor D on X.

Proof of Step 5. We take a general member A of |H| and con-sider the exact sequence

0→ F ⊗OX(−A)→ F → FA → 0.

Since dim SuppFA < dimX, we can find m(FA, H|A) such that

H i(A,FA ⊗OA(mH +D)) = 0

for all i > 0 and m ≥ m(FA, H|A) by induction. Therefore,

H i(X,F ⊗OX((m− 1)H +D)) = H i(X,F ⊗OX(mH +D))

for every i ≥ 2 and m ≥ m(FA, H|A). By Serre’s vanishing theorem,we obtain

H i(X,F ⊗OX((m− 1)H +D)) = 0

for every i ≥ 2 and m ≥ m(FA, H|A).

3.8. FUJITA VANISHING THEOREM 71

Step 6. We can reduce the proof to the case where F = OX .

Proof of Step 6. We assume that Theorem 3.8.1 holds for F =OX . There is an injective homomorphism

α : OX → F ⊗OX(aH)

for some large integer a. We consider the exact sequence

0→ OX → F ⊗OX(aH)→ Cokerα→ 0

and use the induction on rankF . Then we can find m(F , H).

From now on, we assume F = OX .

Step 7. If the characteristic of k is zero, then Theorem 3.8.1 holds.

Proof of Step 7. Let f : Y → X be a resolution. Then weobtain the following exact sequence

0→ f∗ωY → OX(bH)→ C → 0

for some integer b, where dim Supp C < dimX. Note that f∗ωY istorsion-free and rankf∗ωY is one. On the other hand,

Hj(X, f∗ωY ⊗OX(mH +D)) = 0

for every m > 0 and j > 0 by Kollar’s vanishing theorem (see Theorem3.6.3). Therefore,

Hj(X,OX((b+m)H +D)) = 0

for every positive integer m ≥ m(C, H) and j > 0.

If we do not like to use Kollar’s vanishing theorem (see Theorem3.6.3) in Step 7, which was not proved in Section 3.6, then we can usethe following easy lemma.

Lemma 3.8.3. Let X be an irreducible proper variety and let L bea nef and big line bundle on X. Let f : Y → X be a resolution ofsingularities. Then H i(X, f∗ωY ⊗ L) = 0 for every i > 0.

Proof. By the Grauert–Riemenschneider vanishing theorem: The-orem 3.2.7, we have H i(X, f∗ωY ⊗L) ' H i(Y, ωY ⊗f ∗L) for every i. Bythe Kawamata–Viehweg vanishing theorem: Theorem 3.2.1, we obtainH i(Y, ωY ⊗ f ∗L) = 0 for every i > 0. Therefore, we obtain the desiredvanishing theorem.

Step 8. We can reduce the proof to the case where F = ωX , whereωX is the dualizing sheaf of X.

72 3. CLASSICAL VANISHING THEOREMS AND SOME APPLICATIONS

Remark 3.8.4. The dualizing sheaf ωX is denoted by ωX in [Har4,Chapter III §7]. We know that ωX ' ExtN−dimX

OPN(OX , ωPN ) when X ⊂

PN . For details, see the proof of Proposition 7.5 in [Har4, Chapter III§7].

Proof of Step 8. We assume that Theorem 3.8.1 holds for F =ωX . There is an injective homomorphism

β : ωX → OX(cH)

for some positive integer c. Note that ωX is torsion-free. We considerthe exact sequence

0→ ωX → OX(cH)→ Cokerβ → 0.

We note that dim Supp Cokerβ < dimX because

rankωX = rankOX(cH) = 1.

Therefore, we can find m(OX , H) by induction on the dimension andTheorem 3.8.1 for ωX .

From now on, we assume that F = ωX and that the characteristicof k is positive.

Step 9. Theorem 3.8.1 holds when the characteristic of k is posi-tive.

Proof of Step 9. Let X → PN be the embedding induced by H.Let

XF //

X

PN

F// PN

be the commutative diagram of the Frobenius morphisms. By takingRHomOPN

( , ω•PN ) to OX → F∗OX , we obtain

RHomOPN(F∗OX , ω•PN )→ RHomOPN

(OX , ω•PN ).

By Grothendieck duality (see [Har1] and [Con]),

RHomOPN(F∗OX , ω•PN ) ' F∗RHomOPN

(OX , ω•PN ).

Therefore, we obtainγ : F∗ωX → ωX .

Note that ωX = ExtN−dimXOPN

(OX , ωPN ). Let U be a non-empty Zariski

open set of X such that U is smooth. We can easily check that

γ : F∗ωX → ωX

3.8. FUJITA VANISHING THEOREM 73

is surjective on U . Note that the cokernel A of OX → F∗OX is locallyfree on U . Then ExtkOPN

(A, ωPN ) = 0 for k > N − dimX on U . We

consider the exact sequences

0→ Kerγ → F∗ωX → Imγ → 0

and0→ Imγ → ωX → C → 0.

Then dim Supp C < dimX. Note that there is an integer m1 such that

H2(X,Kerγ ⊗OX(mH +D)) = 0

for every m ≥ m1 by Step 5. By applying induction on the dimensionto C, we obtain some positive integer m0 such that

H1(X,F∗ωX ⊗OX(mH +D))→ H1(X,ωX ⊗OX(mH +D))

is surjective for every m ≥ m0. We note that

H1(X,F∗ωX ⊗OX(mH +D)) ' H1(X,ωX ⊗OX(p(mH +D)))

by the projection formula, where p is the characteristic of k. By re-peating the above process, we obtain that

H1(X,ωX ⊗OX(pe(mH +D)))→ H1(X,ωX ⊗OX(mH +D))

is surjective for every e > 0 and m ≥ m0. Note that m0 is independentof the nef divisorD. Therefore, by Serre’s vanishing theorem, we obtain

H1(X,ωX ⊗OX(mH +D)) = 0

for every m ≥ m0. We finish the proof of Theorem 3.8.1. In Step 9, we can use the following elementary lemma to construct

a generically surjective homomorphism F∗ωX → ωX .

Lemma 3.8.5 (see [Ft2, (5.7) Corollary]). Let f : V → W be aprojective surjective morphism between projective varieties defined overan algebraically closed field k with dimV = dimW = n. Then there isa generically surjective homomorphism ϕ : f∗ωV → ωW .

Proof. By definition (see [Har4, Chapter III §7]), Hn(V, ωV ) 6= 0.We consider the Leray spectral sequence

Ep,q2 = Hp(W,Rqf∗ωW )⇒ Hp+q(V, ωV ).

Note that SuppRqf∗ωV is contained in the set

Wq := w ∈ W | dim f−1(w) ≥ q.Since dim f−1(Wq) < n for every q > 0, we have dimWq < n − qfor every q > 0. Therefore, En−q,q

2 = 0 unless q = 0. Thus we obtain

74 3. CLASSICAL VANISHING THEOREMS AND SOME APPLICATIONS

En,02 = Hn(W, f∗ωV ) 6= 0 since Hn(V, ωV ) 6= 0. By the definition of ωW ,

Hom(f∗ωV , ωW ) 6= 0. We take a non-zero element ϕ ∈ Hom(f∗ωV , ωW )and consider Im(ϕ) ⊂ ωW . Since Hom(Im(ϕ), ωW ) 6= 0, we haveHn(W, Im(ϕ)) 6= 0 (see [Har4, Chapter III §7]). This implies thatdim Supp Im(ϕ) = n. Therefore, ϕ : f∗ωV → ωW is generically surjec-tive since rankωW = 1.

Remark 3.8.6. In Lemma 3.8.5, if Rqf∗ωV = 0 for every q > 0,then we obtain Hn(W, f∗ωV ) ' Hn(V, ωV ). We note that Hn(V, ωV ) 'k since k is algebraically closed. Therefore, Hom(f∗ωV , ωW ) ' k. Thismeans that, for any nontrivial homomorphism ψ : f∗ωV → ωW , thereis some a ∈ k \0 such that ψ = aϕ, where ϕ is given in Lemma 3.8.5.Note that Rqf∗ωV = 0 for every q > 0 if f is finite. We also note thatRqf∗ωV = 0 for every q > 0 if the characteristic of k is zero and V hasonly rational singularities by the Grauert–Riemenschneider vanishingtheorem (see Theorem 3.2.7) or by Kollar’s torsion-free theorem: The-orem 3.6.3 (see also Lemma 3.8.7 below).

Although the following lemma is a special case of Kollar’s torsion-freeness (see Theorem 3.6.3), it easily follows from the Kawamata–Viehweg vanishing theorem (see Theorem 3.2.1).

Lemma 3.8.7 (cf. [Ft2, (4.13) Proposition]). Let f : V → W be aprojective surjective morphism from a smooth projective variety V to aprojective variety W , which is defined over an algebraically closed field kof characteristic zero. Then Rqf∗ωV = 0 for every q > dimV −dimW .

Proof. Let A be a sufficiently ample Cartier divisor on W suchthat

H0(W,Rqf∗ωV ⊗OW (A)) ' Hq(V, ωV ⊗OV (f∗A))

and that Rqf∗ωV ⊗ OW (A) is generated by global sections for everyq. We note that the numerical dimension ν(V, f ∗A) of f ∗A is dimW .Therefore, we obtain

Hq(V, ωV ⊗OV (f∗A)) = 0

for q > dimV −dimW = dimV −ν(V, f ∗A) by the Kawamata–Viehwegvanishing theorem: Theorem 3.3.7. Thus, we obtain Rqf∗ωV = 0 forq > dimV − dimW .

Remark 3.8.8. In [Ft2, Section 4], Takao Fujita proves Lemma3.8.7 for a proper surjective morphism f : V → W from a complexmanifold V in Fujiki’s class C to a projective variety W . His proof usesthe theory of harmonic forms. For the details, see [Ft2, Section 4].See also Theorem 3.8.9 below. Note that [Ft2, (4.12) Conjecture] wascompletely solved by Kensho Takegoshi (see [Take1]). See also [F31].

3.8. FUJITA VANISHING THEOREM 75

The following theorem is a weak generalization of Kodaira’s vanish-ing theorem: Theorem 3.7.4. We need no new ideas to prove Theorem3.8.9. The proof of Kodaira’s vanishing theorem based on Bochner’smethod works.

Theorem 3.8.9 (A weak generalization of Kodaira’s vanishing the-orem). Let X be an n-dimensional compact Kahler manifold and let Lbe a line bundle on X whose curvature form

√−1Θ(L) is semi-positive

and has at least k positive eigenvalues on a dense open subset of X.Then H i(X,ωX ⊗ L) = 0 for i > n− k.

We note that H i(X,ωX ⊗ L) is isomorphic to Hn,i(X,L), whichis the space of L-valued harmonic (n, i)-forms on X. By Nakano’sformula, we can easily check that Hn,i(X,L) = 0 for i+ k ≥ n+ 1.

We close this section with a slight generalization of Kollar’s result(cf. [Ko2, Proposition 7.6]), which is related to Lemma 3.8.5. For arelated result, see also [FF, Theorem 7.5].

Proposition 3.8.10. Let f : V → W be a proper surjective mor-phism between normal algebraic varieties with connected fibers, whichis defined over an algebraically closed field k of characteristic zero. As-sume that V and W have only rational singularities. Then Rdf∗ωV 'ωW where d = dimV − dimW .

Proof. We can construct a commutative diagram

Xπ //

g

V

f

Y p// W

with the following properties.

(i) X and Y are smooth algebraic varieties.(ii) π and p are projective birational.(iii) g is projective, and smooth outside a simple normal crossing

divisor Σ on Y .

We note that Rjg∗ωX is locally free for every j (see, for example, [Ko3,Theorem 2.6]). By Grothendieck duality, we have

Rg∗OX ' RHomOY(Rg∗ω

•X , ω

•Y ).

Therefore, we have

OY ' HomOY(Rdg∗ωX , ωY ).

Thus, we obtain Rdg∗ωX ' ωY . By applying p∗, we have

p∗Rdg∗ωX ' p∗ωY ' ωW .

76 3. CLASSICAL VANISHING THEOREMS AND SOME APPLICATIONS

We note that p∗Rdg∗ωX ' Rd(pg)∗ωX since Rip∗R

dg∗ωX = 0 for everyi > 0 (see, for example, [Ko2, Theorem 3.8 (i)], [Ko3, Theorem 2.14,Theorem 3.4 (iii)], or Theorem 5.7.3 (ii) below). On the other hand,

Rd(p g)∗ωX ' Rd(f π)∗ωX ' Rdf∗ωV

since Riπ∗ωX = 0 for every i > 0 and π∗ωX ' ωV . Therefore, we obtainRdf∗ωV ' ωW .

3.9. Applications of Fujita vanishing theorem

In this section, we discuss some applications of Theorem 3.8.1. Formore general statements and other applications, see [Ft2, Section 6].

Theorem 3.9.1 (cf. [Ft1, Theorem (4)] and [Ft2, (6.2) Theorem]).Let F be a coherent sheaf on a scheme X which is proper over analgebraically closed field k. Let L be a nef line bundle on X. Then

dimHq(X,F ⊗ L⊗t) ≤ O(tm−q)

where m = dim SuppF .

Proof. First, we assume that X is projective. We use inductionon q. We put q = 0. Let H be an effective ample Cartier divisor on Xsuch that L ⊗OX(H) is ample. Since

H0(X,F ⊗ L⊗t) ⊂ H0(X,F ⊗ L⊗t ⊗OX(tH))

for every positive integer t, we may assume that L is ample by replacingL with L⊗OX(H). In this case, dimH0(X,F⊗L⊗t) ≤ O(tm) because

dimH0(X,F ⊗ L⊗t) = χ(X,F ⊗ L⊗t)for every t 0 by Serre’s vanishing theorem. When q > 0, by Theorem3.8.1, we have a very ample Cartier divisor A on X such that

Hq(X,F ⊗OX(A)⊗ L⊗t) = 0

for every t ≥ 0. Let D be a general member of |A| such that theinduced homomorphism α : F ⊗OX(−D)→ F is injective. Then

dimHq(X,F ⊗ L⊗t) ≤ dimHq−1(D,Coker(α)⊗OD(A)⊗ L⊗t)≤ O(tm−q)

by the induction hypothesis. Therefore, we obtain the theorem whenX is projective.

Next, we consider the general case. We use the noetherian inductionon SuppF . By the same arguments as in Step 2 and Step 3 in the proofof Theorem 3.8.1, we may assume that X = SuppF is a variety, that is,X is reduced and irreducible. By Chow’s lemma, there is a birationalmorphism f : V → X from a projective variety V . We put G = f ∗F

3.9. APPLICATIONS OF FUJITA VANISHING THEOREM 77

and consider the natural homomorphism β : F → f∗G. Since β is anisomorphism on a non-empty Zariski open subset of X. We considerthe following short exact sequences

0→ Ker(β)→ F → Im(β)→ 0

and

0→ Im(β)→ f∗G → Coker(β)→ 0.

By induction, we obtain

dimHq(X,Ker(β)⊗ L⊗t) ≤ O(tm−q)

and

dimHq−1(X,Coker(β)⊗ L⊗t) ≤ O(tm−q).

Therefore, it is sufficient to see that

dimHq(X, f∗G ⊗ L⊗t) ≤ O(tm−q).

We consider the Leray spectral sequence

Ei,j2 = H i(X,Rjf∗G ⊗ L⊗t)⇒ H i+j(V,G ⊗ (f ∗L)⊗t).

Then we have

dimHq(X, f∗G ⊗ L⊗t) ≤∑j≥1

dimHq−j−1(X,Rjf∗G ⊗ L⊗t)

+ dimHq(V,G ⊗ (f∗L)⊗t).

Note that

dimHq(V,G ⊗ (f ∗L)⊗t) ≤ O(tm−q)

since V is projective. On the other hand, we have

dim SuppRjf∗G ≤ dimX − j − 1

for every j ≥ 1 as in the proof of Lemma 3.8.5. Therefore,

dimHq−j−1(X,Rjf∗G ⊗ L⊗t) ≤ O(tm−q)

by the induction hypothesis. Thus, we obtain

dimHq(X,F ⊗ L⊗t) ≤ O(tm−q).

We complete the proof.

As an application of Theorem 3.9.1, we can prove Fujita’s numericalcharacterization of nef and big line bundles. We note that the charac-teristic of the base field is arbitrary in Corollary 3.9.2. Corollary 3.9.2in characteristic zero, which is due to Sommese, is well known. See, forexample, [Ka3, Lemma 3].

78 3. CLASSICAL VANISHING THEOREMS AND SOME APPLICATIONS

Corollary 3.9.2 (cf. [Ft1, Theorem (6)] and [Ft2, (6.5) Corol-lary]). Let L be a nef line bundle on a proper algebraic irreducible vari-ety V defined over an algebraically closed field k with dimV = n. ThenL is big if and only if the self-intersection number Ln is positive.

Proof. Let ν : Xν → X be the normalization. By replacing Xand L with Xν and ν∗L, we may assume that X is normal. It is wellknown that

χ(V,L⊗t)− Ln

n!tn ≤ O(tn−1).

By Theorem 3.9.1, we have

dimH0(V,L⊗t)− χ(V,L⊗t) ≤ O(tn−1).

Therefore, L is big if and only if Ln > 0. Note that Ln ≥ 0 since L isnef.

Corollary 3.9.3 (cf. [Ft1, Corollary (7)] and [Ft2, (6.7) Corol-lary]). Let L be a nef and big line bundle on a projective irreduciblevariety V defined over an algebraically closed field k with dimV = n.Then, for any coherent sheaf F on V , we have

dimHq(V,F ⊗ L⊗t) ≤ O(tn−q−1)

for every q ≥ 1. In particular, Hn(V,F ⊗ L⊗t) = 0 for every t 0.

Proof. Let A be an ample Cartier divisor such that

H i(V,F ⊗OV (A)⊗ L⊗t) = 0

for every i > 0 and t ≥ 0 (see Theorem 3.8.1). Since L is big, there is apositive integer m such that |L⊗m⊗OV (−A)| 6= ∅ by Kodaira’s lemma(see Lemma 2.1.18). We take D ∈ |L⊗m ⊗OV (−A)| and consider thehomomorphism γ : F ⊗OV (−D)→ F induced by γ. Then we have

dimHq(V,F ⊗ L⊗t) ≤ dimHq(V,Coker(γ)⊗ L⊗t)+ dimHq(V, Im(γ)⊗ L⊗t),

and

dimHq(V, Im(γ)⊗ L⊗t) ≤ dimHq(V,F ⊗OV (−D)⊗ L⊗t)+ dimHq+1(V,Ker(γ)⊗ L⊗t)

= dimHq+1(V,Ker(γ)⊗ L⊗t)for every t ≥ m. This is because

Hq(V,F ⊗OV (−D)⊗ L⊗t)' Hq(V,F ⊗OV (A)⊗ L⊗(t−m)) = 0

3.10. TANAKA VANISHING THEOREMS 79

for every t ≥ m. Note that

dimHq(V,Coker(γ)⊗ L⊗t) ≤ O(tn−1−q)

by Theorem 3.9.1 since Supp Coker(γ) is contained in D. On the otherhand,

dimHq+1(V,Ker(γ)⊗ L⊗t) ≤ O(tn−q−1)

by Theorem 3.9.1. By combining there estimates, we obtain the desiredestimate.

3.10. Tanaka vanishing theorems

In this section, we discuss Tanaka’s vanishing theorems. It is wellknown that Kodaira’s vanishing theorem does not always hold evenfor surfaces when the characteristic of the base filed is positive. In[Tana2], Hiromu Tanaka obtained the following vanishing theorem asan application of Fujita’s vanishing theorem: Theorem 3.8.1. They aresufficient for X-method for surfaces in positive characteristic.

Theorem 3.10.1 (Kodaira type vanishing theorem). Let X be asmooth projective surface defined over an algebraically closed filed k ofpositive characteristic. Let A be an ample Cartier divisor on X and letN be a nef Cartier divisor on X which is not numerically trivial. Then

H i(X,OX(KX + A+mN)) = 0

for every i > 0 and every m 0.

More generally, Hiromu Tanaka proved the following vanishing the-orems.

Theorem 3.10.2 (Kawamata–Viehweg type vanishing theorem).Let X be a smooth projective surface defined over an algebraically closedfiled k of positive characteristic. Let A be an ample R-divisor on Xwhose fractional part has a simple normal crossing support and let Nbe a nef Cartier divisor on X which is not numerically trivial. Then

H i(X,OX(KX + dAe+mN)) = 0

for every i > 0 and every m 0.

Theorem 3.10.3 (Nadel type vanishing theorem). Let X be a nor-mal projective surface defined over an algebraically closed field k ofpositive characteristic. Let ∆ be an R-divisor on X such that KX + ∆is R-Cartier. Let N be a nef Cartier divisor on X which is not numer-ically trivial. Let L be a Cartier divisor on X such that L− (KX + ∆)is nef and big. Then

H i(X,OX(L+mN)⊗ J (X,∆)) = 0

80 3. CLASSICAL VANISHING THEOREMS AND SOME APPLICATIONS

for every i > 0 and every m 0, where J (X,∆) is the multiplier idealsheaf of the pair (X,∆).

For the details of Theorems 3.10.1, 3.10.2, and 3.10.3, and somerelated topics, see [Tana2].

3.11. Ambro vanishing theorem

In this section, we prove Ambro’s vanishing theorem in [Am2],which is an application of Corollary 3.1.2.

Theorem 3.11.1. Let X be a smooth projective variety and let ∆be a reduced simple normal crossing divisor on X. Assume that X \∆is contained in an affine Zariski open set U of X. Then we have

H i(X,OX(KX + ∆)) = 0

for every i > 0.

Our proof of Theorem 3.11.1 is based on Corollary 3.1.2 and theweak factorization theorem in [AKMW].

Proof. We may assume that U ⊂ Cn. By taking the closure of Uin Pn and taking some suitable blow-ups outside U , we can construct asmooth projective variety X ′ with the following properties (cf. Good-man’s criterion in [Har3, Chapter II Theorem 6.1]).

(i) U ⊂ X ′ and Σ = X ′ \U is a simple normal crossing divisor onX ′.

(ii) There is a simple normal crossing divisor ∆′ on X ′ such thatΣ ≤ ∆′ and that (X ′,∆′)|U = (X,∆)|U . In particular, X ′ \∆′ = X \∆.

(iii) There is an effective ample Cartier divisor D′ on X ′ such thatSuppD′ ⊂ Supp ∆′.

By Corollary 3.1.2, we obtain that

H i(X ′,OX′(KX′ + ∆′)) = 0

for every i > 0.

Claim. We have

H i(X,OX(KX + ∆)) ' H i(X ′,OX′(KX′ + ∆′))

for every i.

Proof of Claim. By the weak factorization theorem (see [AKMW,Theorem 0.3.1]), we may assume that f : X ′ → X is a blow-up whose

3.11. AMBRO VANISHING THEOREM 81

center C, which is contained in ∆, is smooth and has simple normalcrossings with ∆. It is sufficient to check that

Rjf∗OX′(KX′ + ∆′) = 0

for every j > 0 and

f∗OX′(KX′ + ∆′) ' OX(KX + ∆).

By shrinking X ′, we may assume that C is irreducible. Then we have

KX′ + ∆′ = f ∗(KX + ∆) + (c−m)E

and

KX′ = f ∗KX + (c− 1)E

where c = codimXC, m = multC∆, and E is the exceptional divisor off . Since c−m ≥ 0 and E is f -exceptional, we obtain

f∗OX′(KX′ + ∆′) ' OX(KX + ∆).

Since

KX′ + ∆′ −KX′ ∼f (1−m)E

is f -nef and f -big,

Rjf∗OX′(KX′ + ∆′) = 0

for every j > 0 by the Kawamata–Viehweg vanishing theorem: Theo-rem 3.2.1. Of course, we can directly check the above vanishing state-ment because f : X ′ → X is a blow-up whose center is smooth.

Therefore, we obtain the desired vanishing theorem.

We learned the following example from Takeshi Abe.

Example 3.11.2. There is a projective birational morphism f :X → Y from a smooth projective variety X to a normal projectivevariety Y with the following properties.

(i) The exceptional locus Exc(f) of f is an irreducible curve C onX.

(ii) There is a prime Weil divisor H on Y with P := f(C) ∈ Hwhich is an ample Cartier divisor on Y .

Then U := X \ f−1(H) ' Y \ H is an affine Zariski open set of X.In this case, D := X \ U is a prime Weil divisor on X which is a nefand big Cartier divisor on X such that D ·C = 0. Therefore, D is notample. Note that we can choose f : X → Y to be a three-dimensionalflopping contraction.

For some related results, see [Har3, Chapter II].

82 3. CLASSICAL VANISHING THEOREMS AND SOME APPLICATIONS

3.12. Kovacs’s characterization of rational singularities

In this section, we discuss Kovacs’s characterization of rational sin-gularities in [Kv3].

Let us recall the definition of rational singularities.

Definition 3.12.1 (Rational singularities). Let X be a variety. Ifthere exists a resolution of singularities f : Y → X such that Rif∗OY =0 for every i > 0 and f∗OY ' OX , equivalently, the natural mapOX → Rf∗OY is a quasi-isomorphism, then X is said to have onlyrational singularities.

Lemma 3.12.2 ([KKMS]). Let X be a variety and let f : Y → Xbe a resolution of singularities. Then the natural map OX → Rf∗OY isa quasi-isomorphism if and only if X is Cohen–Macaulay and f∗ωY 'ωX .

Proof. Assume that X is Cohen–Macaulay and f∗ωY ' ωX . ThenRf∗ω

•Y ' ω•X by the Grauert–Riemenschneider vanishing theorem: The-

orem 3.2.7. By Grothendieck duality, we obtain

OX ' RHom(ω•X , ω•X) ' RHom(Rf∗ω

•Y , ω

•X)

' Rf∗RHom(ω•Y , ω•Y ) ' Rf∗OY .

Assume that the natural mapOX → Rf∗OY is a quasi-isomorphism.By Grothendieck duality,

Rf∗ω•Y ' RHom(Rf∗OY , ω•X)

' RHom(OX , ω•X) ' ω•X .

Note that ω•Y ' ωY [d] where d = dimX = dimY . Then hi(ω•X) =Ri+dωY = 0 for i > −d by the Grauert–Riemenschneider vanishingtheorem: Theorem 3.2.7. Therefore, X is Cohen–Macaulay and ω•X 'ωX [d]. Thus we obtain f∗ωY ' ωX .

We can easily check the following lemmas.

Lemma 3.12.3. Let X be a smooth variety. Then X has only ra-tional singularities.

Proof. Note that the identity map idX : X → X is a resolutionof singularities of X since X itself is smooth.

Lemma 3.12.4. Assume that X has only rational singularities. Letf : Y → X be any resolution of singularities. Then Rif∗OY = 0 forevery i > 0 and f∗OY ' OX .

3.12. KOVACS’S CHARACTERIZATION OF RATIONAL SINGULARITIES 83

Proof. By Lemma 3.12.2, it is sufficient to see f∗ωY ' ωX . On theother hand, it is easy to see that f∗ωY is independent of the resolutionf : Y → X. Therefore, we have f∗ωY ' ωX by Lemma 3.12.2 andDefinition 3.12.1.

The following theorem is Kovacs’s characterization of rational sin-gularities.

Theorem 3.12.5 ([Kv3, Theorem 1]). Let f : Y → X be a mor-phism between varieties and let α : OX → Rf∗OY be the associatednatural morphism. Assume that Y has only rational singularities andthere exists a morphism β : Rf∗OY → OX such that β α is a quasi-isomorphism in the derived category. Then X has only rational singu-larities.

Proof. We construct the following commutative diagram:

Y

ef

σ // Y

f

X π

// X

such that σ and π are resolutions of singularities. Then we have thefollowing commutative diagram:

OXa

α // Rf∗OYb

Rπ∗O eX c// Rf∗Rσ∗OeY .

Note that b is a quasi-isomorphism because Y has only rational singu-larities. Therefore,

(β b−1 c) a : OX → Rπ∗O eX → OXis a quasi-isomorphism. Thus, we may assume that f is a resolution ofsingularities. We apply RHom( , ω•X) to

OXα−→ Rf∗OY

β−→ OX .By Grothendieck duality, we obtain

ω•Xα∗←− Rf∗ω

•Y

β∗←− ω•X

such that α∗ β∗ is a quasi-isomorphism. Therefore, we obtain

hi(ω•X) ⊂ Rif∗ω•Y ' Ri+df∗ωY

where d = dimX = dimY . By the Grauert–Riemenschneider vanish-ing theorem: Theorem 3.2.7, we have Ri+dωY = 0 for every i > −d.

84 3. CLASSICAL VANISHING THEOREMS AND SOME APPLICATIONS

This implies that hi(ω•X) = 0 for every i > −d. This means that X isCohen–Macaulay. By the above argument, we obtain

ωXh−d(β∗)

// f∗ωYh−d(α∗)

// ωX

such that the composition is an isomorphism. This implies f∗ωY ' ωX .Therefore, X has only rational singularities by Lemma 3.12.2.

The arguments in the proof of Theorem 3.12.5 is very useful forvarious applications (see the proof of Theorem 3.13.6).

We close this section with a well-known vanishing theorem for va-rieties with only rational singularities. It is an easy application of theKawamata–Viehweg vanishing theorem.

Theorem 3.12.6 (Vanishing theorem for varieties with only ratio-nal singularities). Let X be a normal complete variety with only rationalsingularities. Let D be a nef and big Cartier divisor on X. Then

H i(X,ωX ⊗OX(D)) = 0

for i > 0, equivalently, by Serre duality,

H i(X,OX(−D)) = 0

for i < dimX.

Proof. Let f : Y → X be a resolution of singularities. Since Xhas only rational singularities,

H i(X,OX(−D)) ' H i(Y,OY (−f ∗D))

for every i. Since f is birational, f ∗D is nef and big. Therefore,H i(Y,OY (−f ∗D)) = 0 for every i < dimY = dimX by the Kawamata–Viehweg vanishing theorem: Theorem 3.2.1. Note that H i(X,ωX ⊗OX(D)) is dual to HdimX−i(X,OX(−D)) by Serre duality because Xis Cohen–Macaulay.

3.13. Basic properties of dlt pairs

In this section, we prove some basic properties of dlt pairs. We notethat the notion of dlt pairs plays very important roles in the recentdevelopments of the minimal model program after [KoMo]. We alsonote that the notion of dlt pairs was introduced by Shokurov [Sh2].

First, let us prove the following well-known theorem.

Theorem 3.13.1. Let (X,D) be a dlt pair. Then X has only ra-tional singularities.

For the proof of Theorem 3.13.1, the following formulation of theKawamata–Viehweg vanishing theorem is useful.

3.13. BASIC PROPERTIES OF DLT PAIRS 85

Theorem 3.13.2 (Kawamata–Viehweg vanishing theorem). Let f :Y → X be a projective surjective morphism onto a variety Y and letM be a Cartier divisor on Y . Let ∆ be a boundary R-divisor on Ysuch that Supp ∆ is a normal crossing divisor on Y . Assume thatM − (KY + ∆) is f -ample. Then

Rif∗OY (M) = 0

for every i > 0.

It is obvious that Theorem 3.13.2 contains Norimatsu’s vanishingtheorem: Theorem 3.2.12.

Proof of Theorem 3.13.2. We put D = M − (KY + (1− ε)∆)for some small positive number ε. Then D is an f -ample R-divisor onY such that dDe = M −KY and that SuppD is a normal crossingdivisor on Y . By Theorem 3.2.9, we obtain Rif∗OY (KY + dDe) = 0for every i > 0. This means that Rif∗OY (M) = 0 for every i > 0.

Let us give a proof of Theorem 3.13.1 based on Theorem 3.12.5,which was first obtained in [F17, Theorem 4.9]. For a related result,see [Nak2, Chapter VII, 1.1.Theorem].

Proof of Theorem 3.13.1. By the definition of dlt pairs, wecan take a resolution f : Y → X such that Exc(f) and Exc(f) ∪Supp f−1

∗ D are both simple normal crossing divisors on Y and that

KY + f−1∗ D = f ∗(KX +D) + E

with dEe ≥ 0. We can take an effective f -exceptional divisor A onY such −A is f -ample (see, for example, Remark 2.3.18 and [F12,Proposition 3.7.7]). Then

dEe − (KY + f−1∗ D + −E+ εA) = −f ∗(KX +D)− εA

is f -ample for ε > 0. If 0 < ε 1, then f−1∗ D + −E + εA is a

boundary R-divisor whose support is a simple normal crossing divisoron Y . Therefore, Rif∗OY (dEe) = 0 for i > 0 by Theorem 3.13.2 andf∗OY (dEe) ' OX . Note that dEe is effective and f -exceptional. Thus,the composition

OX → Rf∗OY → Rf∗OY (dEe) ' OXis a quasi-isomorphism in the derived category. So, X has only rationalsingularities by Theorem 3.12.5.

Remark 3.13.3. It is curious that Theorem 3.13.1 is missing in[Kv3]. As we saw in the proof of Theorem 3.13.1, it easily followsfrom Kovacs’s characterization of rational singularities (see Theorem

86 3. CLASSICAL VANISHING THEOREMS AND SOME APPLICATIONS

3.12.5 and [Kv3, Theorem 1]). In [Kv3, Theorem 4], Kovacs onlyproved the following statement. Let X be a variety with log terminalsingularities. Then X has only rational singularities.

3.13.4 (Weak log-terminal singularities). The proof of Theorem3.13.1 works for weak log-terminal singularities (see Definition 2.3.21).Thus, we can recover [KMM, Theorem 1-3-6], that is, we obtain thefollowing statement.

Theorem 3.13.5 (see [KMM, Theorem 1-3-6]). All weak log-terminalsingularities are rational.

We do not need the difficult vanishing theorem due to Elkik andFujita (see Theorem 3.14.1) to obtain the above theorem. In Theorem3.13.1, if we assume that (X,D) is only weak log-terminal singularities,then we can not always make Exc(f) and Exc(f) ∪ Supp f−1

∗ D simplenormal crossing divisors. We can only make them normal crossing di-visors. However, Theorem 3.13.2 works in this setting. Thus, the proofof Theorem 3.13.1 works for weak log-terminal singularities. Anyway,the notion of weak log-terminal singularities is not useful in the recentminimal model program.

The following theorem generalizes [Koetal, 17.5 Corollary], whereit was only proved that S is semi-normal and satisfies Serre’s S2 condi-tion. Theorem 3.13.6 was first obtained in [F17] in order to understand[Koetal, 17.5 Corollary].

Theorem 3.13.6 ([F17, Theorem 4.14]). Let X be a normal varietyand let S + B be a boundary R-divisor such that (X,S + B) is dlt, Sis reduced, and bBc = 0. Let S = S1 + · · · + Sk be the irreducibledecomposition. We put T = S1 + · · · + Sl for some l with 1 ≤ l ≤k. Then T is semi-normal, Cohen–Macaulay, and has only Du Boissingularities.

Proof. Let f : Y → X be a resolution of singularities such that

KY + S ′ +B′ = f∗(KX + S +B) + E

with the following properties (see Remark 2.3.18):

(i) S ′ (resp. B′) is the strict transform of S (resp. B).(ii) Supp(S ′+B′)∪Exc(f) and Exc(f) are simple normal crossing

divisors on Y .(iii) f is an isomorphism over the generic point of any log canonical

center of (X,S +B).(iv) dEe ≥ 0.

3.13. BASIC PROPERTIES OF DLT PAIRS 87

We write S = T + U . Let T ′ (resp. U ′) be the strict transform of T(resp. U) on Y . We consider the following short exact sequence

0→ OY (−T ′ + dEe)→ OY (dEe)→ OT ′(dE|T ′e)→ 0.

Since −T ′ + E ∼R,f KY + U ′ +B′ and E ∼R,f KY + S ′ +B′, we have

−T ′ + dEe ∼R,f KY + U ′ +B′ + −Eand

dEe ∼R,f KY + S ′ +B′ + −E.By Theorem 3.2.11, we obtain

Rif∗OY (−T ′ + dEe) = Rif∗OY (dEe) = 0

for every i > 0. Therefore, we have

0→ f∗OY (−T ′ + dEe)→ OX → f∗OT ′(dE|T ′e)→ 0

and Rif∗OT ′(dE|T ′e) = 0 for every i > 0. Note that dEe is effectiveand f -exceptional. Thus we obtain

OT ' f∗OT ′ ' f∗OT ′(dE|T ′e).Since T ′ is a simple normal crossing divisor, T is semi-normal. By theabove vanishing result, we obtain Rf∗OT ′(dE|T ′e) ' OT in the derivedcategory. Therefore, the composition

OT → Rf∗OT ′ → Rf∗OT ′(dE|T ′e) ' OTis a quasi-isomorphism. Apply

RHomT ( , ω•T )

toOT → Rf∗OT ′ → OT .

Then the composition

ω•T → Rf∗ω•T ′ → ω•T

is a quasi-isomorphism by Grothendieck duality. Hence, we have

hi(ω•T ) ⊆ Rif∗ω•T ′ ' Ri+df∗ωT ′ ,

where d = dimT = dimT ′.

Claim (see also Lemma 5.6.1). Rif∗ωT ′ = 0 for every i > 0.

Proof of Claim. We use induction on the number of the irre-ducible components of T ′. If T ′ is irreducible, then Claim follows fromthe Grauert–Riemenschneider vanishing theorem: Theorem 3.2.7. LetS ′i be the strict transform of Si on Y for every i. Let W be any irre-ducible component of S ′i1 ∩ · · · ∩ S

′im for i1, · · · , im ⊂ 1, 2, · · · , k.

Then f : W → f(W ) is birational by the construction of f . Therefore,

88 3. CLASSICAL VANISHING THEOREMS AND SOME APPLICATIONS

every Cartier divisor on W is f -big. We put T ′ = S ′1 +T ′0 and considerthe short exact sequence

0→ OT ′0(−S ′1)→ OT ′ → OS′

1→ 0.

By taking ⊗ωT ′ , we obtain

0→ ωT ′0→ ωT ′ → ωS′

1⊗OS′

1(T ′|S′

1)→ 0.

Then we have the following long exact sequence

· · · → Rif∗ωT ′0→ Rif∗ωT ′ → Rif∗OS′

1(KS′

1+ T ′|S′

1)→ · · · .

By Theorem 3.2.11, Rif∗OS′1(KS′

1+ T ′|S′

1) = 0 for every i > 0. By

induction on the number of the irreducible components, we obtain thatRif∗ωT ′

0= 0 for every i > 0. Therefore, we obtain the desired vanishing

theorem. Therefore, by Claim, hi(ω•T ) = 0 for i 6= −d. Thus, T is Cohen–

Macaulay. This argument is the same as the proof of Theorem 3.12.5.Since T ′ is a simple normal crossing divisor, T ′ has only Du Bois sin-gularities (see, for example, Lemma 5.3.8). Note that the composition

OT → Rf∗OT ′ → OTis a quasi-isomorphism. It implies that T has only Du Bois singularities(see [Kv1, Corollary 2.4]). Since the composition

ωT → f∗ωT ′ → ωT

is an isomorphism, we obtain f∗ωT ′ ' ωT . By Grothendieck duality,

Rf∗OT ′ ' RHomT (Rf∗ω•T ′ , ω•T ) ' RHomT (ω•T , ω

•T ) ' OT .

So, we have Rif∗OT ′ = 0 for every i > 0. We obtained the following vanishing theorem in the proof of Theo-

rem 3.13.6.

Corollary 3.13.7. Under the notation in the proof of Theorem3.13.6, Rif∗OT ′ = 0 for every i > 0 and f∗OT ′ ' OT .

As a special case, we have:

Corollary 3.13.8 ([KoMo, Corollary 5.52]). Let (X,S+B) be adlt pair as in Theorem 3.13.6. Then Si is normal for every i.

Proof. We put T = Si. Then Si is normal since f∗OT ′ ' OT (seeCorollary 3.13.7).

Let us discuss a nontrivial example. This example shows the sub-tleties of the notion of dlt pairs.

3.13. BASIC PROPERTIES OF DLT PAIRS 89

Example 3.13.9 (cf. [KMM, Remark 0-2-11. (4)]). We considerthe P2-bundle

π : V = PP2(OP2 ⊕OP2(1)⊕OP2(1))→ P2.

Let F1 = PP2(OP2 ⊕ OP2(1)) and F2 = PP2(OP2 ⊕ OP2(1)) be two hy-persurfaces of V which correspond to projections

OP2 ⊕OP2(1)⊕OP2(1)→ OP2 ⊕OP2(1)

given by (x, y, z) 7→ (x, y) and (x, y, z) 7→ (x, z). Let Φ : V → W be theflipping contraction that contracts the negative section of π : V → P2,that is, the section corresponding to the projection

OP2 ⊕OP2(1)⊕OP2(1)→ OP2 → 0.

Let C ⊂ P2 be an elliptic curve. We put Y = π−1(C), D1 = F1|Y , andD2 = F2|Y . Let f : Y → X be the Stein factorization of Φ|Y : Y →Φ(Y ). Then the exceptional locus of f is E = D1 ∩D2. We note thatY is smooth, D1 +D2 is a simple normal crossing divisor, and E ' Cis an elliptic curve. Let g : Z → Y be the blow-up along E. Then

KZ +D′1 +D′2 +D = g∗(KY +D1 +D2),

where D′1 (resp. D′2) is the strict transform of D1 (resp. D2) and D isthe exceptional divisor of g. Note that D ' C × P1. Since

−D + (KZ +D′1 +D′2 +D)− (KZ +D′1 +D′2) = 0,

we obtain that Rif∗(g∗OZ(−D + KZ + D′1 + D′2 + D)) = 0 for everyi > 0 by Theorem 5.7.3 below. We note that f g is an isomorphismoutside D. We consider the following short exact sequence

0→ IE → OY → OE → 0,

where IE is the defining ideal sheaf of E. Since IE = g∗OZ(−D), weobtain that

0→ f∗(IE ⊗OY (KY +D1 +D2))→ f∗OY (KY +D1 +D2)

→ f∗OE(KY +D1 +D2)→ 0

by R1f∗(IE ⊗OY (KY +D1 +D2)) = 0. By adjunction,

OE(KY +D1 +D2) ' OE.Therefore, OY (KY +D1 +D2) is f -free. In particular,

KY +D1 +D2 = f ∗(KX +B1 +B2),

where B1 = f∗D1 and B2 = f∗D2. Thus, −D− (KZ +D′1 +D′2) ∼fg 0.So, we have

Rif∗IE = Rif∗(g∗OZ(−D)) = 0

90 3. CLASSICAL VANISHING THEOREMS AND SOME APPLICATIONS

for every i > 0 by Theorem 5.7.3 below. This implies that Rif∗OY 'Rif∗OE for every i > 0. Thus, R1f∗OY ' C(P ), where P = f(E). Weconsider the following spectral sequence

Ep,q = Hp(X,Rqf∗OY ⊗OX(−mA))⇒ Hp+q(Y,OY (−mA)),

where A is an ample Cartier divisor on X and m is any positive in-teger. Since H1(Y,OY (−mf ∗A)) = H2(Y,OY (−mf ∗A)) = 0 by theKawamata–Viehweg vanishing theorem (see Theorem 3.2.1), we have

H0(X,R1f∗OY ⊗OX(−mA)) ' H2(X,OX(−mA)).

If we assume that X is Cohen–Macaulay, then we have

H2(X,OX(−mA)) = 0

for everym 0 by Serre duality and Serre’s vanishing theorem. On theother hand, H0(X,R1f∗OY ⊗OX(−mA)) ' C(P ) because R1f∗OY 'C(P ). This is a contradiction. Thus, X is not Cohen–Macaulay. Inparticular, (X,B1 + B2) is log canonical but not dlt. We note thatExc(f) = E is not a divisor on Y .

Let us recall that Φ : V → W is a flipping contraction. Let Φ+ :V + → W be the flip of Φ. We can check that

V + = PP1(OP1 ⊕OP1(1)⊕OP1(1)⊕OP1(1))

and the flipped curve E+ ' P1 is the negative section of π+ : V + → P1,that is, the section corresponding to the projection

OP1 ⊕OP1(1)⊕OP1(1)⊕OP1(1)→ OP1 → 0.

Let Y + be the strict transform of Y on V +. Then Y + is Gorenstein, logcanonical along E+ ⊂ Y +, and smooth outside E+. Let D+

1 (resp. D+2 )

be the strict transform of D1 (resp. D2) on Y +. If we take a Cartierdivisor B on Y suitably, then

(Y,D1 +D2) //_______

f%%LLLLLLLLLLL

(Y +, D+1 +D+

2 )

xxpppppppppppp

X

is the B-flop of f : Y → X. In this example, the flopping curve E isa smooth elliptic curve and the flopped curve E+ is P1. We note that(Y,D1 +D2) is dlt. However, (Y +, D+

1 +D+2 ) is log canonical but not

dlt.

We close this section with Kovacs’s vanishing theorem.

3.14. ELKIK–FUJITA VANISHING THEOREM 91

Theorem 3.13.10 (cf. [Kv5, Theorem 1.2] and [F34, Theorem 1]).Let (X,∆) be a log canonical pair and let f : Y → X be a proper bira-tional morphism from a smooth variety Y such that Exc(f)∪Supp f−1

∗ ∆is a simple normal crossing divisor on Y . In this situation, we can write

KY = f ∗(KX + ∆) +∑i

aiEi.

We put E =∑

ai=−1Ei. Then we have

Rif∗OY (−E) = 0

for every i > 0.

The proof given in [F34] is essentially the same as the proof ofTheorem 3.13.6 with the aid of the minimal model program. For aslightly simpler proof of Theorem 3.13.10, see [Ch2, Section 4]. Theoriginal proof of Theorem 3.13.10 in [Kv5] uses the notion of Du Boispairs (see Definition 5.3.5) and the minimal model program. Anyway,we do not know any proof without using the minimal model program.So we omit the details here.

Remark 3.13.11. If (X,∆) is klt, then Theorem 3.13.10 says thatX has only rational singularities.

Remark 3.13.12. In [Kv5], Kovacs proved Theorem 3.13.10 underthe extra assumption that X is Q-factorial. When we use Theorem3.13.10 for the study of log canonical singularities, the assumption thatX is Q-factorial is very restrictive. See, for example, [Ch2].

3.14. Elkik–Fujita vanishing theorem

The Elkik–Fujita vanishing theorem (see [Elk] and [Ft3]) is a verydifficult vanishing theorem in [KMM].

Theorem 3.14.1 ([KMM, Theorem 1-3-1]). Let f : Y → X be aprojective birational morphism from a smooth variety Y onto a variety

X. Let L and L be Cartier divisors on Y . Assume that there exist R-divisors D and D on Y and an effective Cartier divisor E on Y suchthat the following conditions are satisfied:

(i) SuppD and Supp D are simple normal crossing divisors, and

bDc = bDc = 0,

(ii) both −L−D and −L− D are f -nef,

(iii) KY ∼ L+ L+ E, and(iv) E is f -exceptional.

Then Rif∗OY (L) = 0 for every i > 0.

92 3. CLASSICAL VANISHING THEOREMS AND SOME APPLICATIONS

We give a very simple proof due to Chih-Chi Chou (see [Ch1]).The original proof in [KMM, §1-3] is much harder than the proofgiven below.

Proof. We consider

α : f∗OY (L) ' τ≤0Rf∗OY (L)→ Rf∗OY (L)

andβ : Rf∗OY (L)→ Rf∗OY (L+ E)

in the derived category of coherent sheaves. Since

L+ E − (KY + D) ∼ −L− Dis f -nef and f -big, we obtain

Rif∗OY (L+ E) = 0

for every i > 0 by the Kawamata–Viehweg vanishing theorem: Theorem3.2.9. By Lemma 3.14.2 below,

f∗OY (L+ E) = f∗OY (L).

Therefore, the composition

β α : f∗OY (L)→ Rf∗OY (L)→ Rf∗OY (L+ E)

is a quasi-isomorphism. By taking RHom( , ω•X), we obtain

RHom(f∗OY (L), ω•X)α∗←− RHom(Rf∗OY (L), ω•X)

β∗←− RHom(Rf∗OY (L+ E), ω•X)

such that α∗ β∗ is a quasi-isomorphism. By Grothendieck duality,

RHom(Rf∗OY (L), ω•X) ' Rf∗OY (KY − L)[n]

and

RHom(Rf∗OY (L+ E), ω•X) ' Rf ∗OY (KY − L− E)[n]

where n = dimX = dimY . Since

KY − L− (KY +D) = −L−Dis f -nef and f -big, we obtain

Rif∗OY (KY − L) = 0

for every i > 0 by the Kawamata–Viehweg vanishing theorem: Theorem3.2.9. Since

α∗ β∗ : Rf∗OY (KY − L− E)[n]→ Rf∗OY (KY − L)[n]

→ RHom(f∗OY (L), ω•X)

3.14. ELKIK–FUJITA VANISHING THEOREM 93

is a quasi-isomorphism, we have

Rif∗OY (KY − L− E) = 0

for every i > 0. This implies that Rif∗OY (L) = 0 for every i > 0 bythe condition (iii). By symmetry, we obtain Rif∗OY (L) = 0 for everyi > 0.

We have already used the following Fujita’s lemma in the proof ofTheorem 3.14.1 (see also [KMM, Lemma 1-3-2]).

Lemma 3.14.2 ([Ft3, (2.2) Lemma]). Let f : Y → X be a projectivebirational morphism from a smooth variety Y onto a variety X, let Lbe a Cartier divisor on Y , let D be an R-Cartier R-divisor on Y , andlet E be a Cartier divisor on Y . Assume that SuppD is a simplenormal crossing divisor, bDc = 0, −L − D is f -nef, and that E iseffective and f -exceptional. Then f∗OE(L + E) = 0. In particular,f∗OY (L) = f∗OY (L+ E).

Proof. For any reduced irreducible component Ej of E, we havethe exact sequence

0→ f∗OE′(L+ E ′)→ f∗OE(L+ E)→ f∗OEj(L+ E),

where E ′ = E −Ej. Thus, by induction on the number of componentsof E, it is sufficient to prove that there exists a reduced irreduciblecomponent E0 of E such that f∗OE0(L + E) = 0. We will prove thisby induction on n = dimX.

First, we assume n = 2. We write E −D = A−B, where A and Bare effective R-divisors without common components. Since bDc = 0,we have A 6= 0. Since SuppA ⊂ SuppE, A is f -exceptional. Therefore,by Lemma 2.3.24, we have A · E0 < 0 for some irreducible componentE0 of A. Sine −L−D is f -nef, we have

(E + L) · E0 ≤ (E −D) · E0 ≤ A · E0 < 0,

which implies f∗OE0(L+ E) = 0.Next, we assume n ≥ 3. We will derive a contradiction assuming

that f∗OEj(L + Ej) 6= 0 for every irreducible component Ej of E. By

replacing X with an arbitrary affine open set of X, we may assumethat X is affine.

If dim f(E) = 0, then

H0(Ej,OEj(L+ E)) 6= 0

for every Ej. We take a general hyperplane section Y ′ of Y . Then

H0(Ej ∩ Y ′,OEj∩Y ′(L+ E)) 6= 0

for every Ej. This is a contradiction by induction hypothesis.

94 3. CLASSICAL VANISHING THEOREMS AND SOME APPLICATIONS

If dim f(E) ≥ 1, then we take a general hyperplane section X ′ ofX and apply induction to f : Y ′ := f−1(X ′) → X ′. Then, we havef∗OEj∩Y ′(L + E) 6= 0 for every Ej with dim f(Ej) ≥ 1, which is acontradiction by induction hypothesis.

Remark 3.14.3. In [KMM], f in Theorem 3.14.1 and Lemma3.14.2 is assumed to be proper. However, we assume that f is pro-jective in Theorem 3.14.1 and Lemma 3.14.2 since we take a generalhyperplane section Y ′ of Y in the proof of Lemma 3.14.2.

The following remark is obvious by the proof of Theorem 3.14.1.

Remark 3.14.4 (see [KMM, Remark 1-3-5]). It is easy to see thatTheorem 3.14.1 holds under the following conditions (a) and (b) insteadof (i) and (ii):

(a) SuppD and Supp D are normal crossing divisors, bDc = 0,

and D is a boundary R-divisor,

(b) −L−D is f -nef and −L− D is f -ample.

We give a proof of [KMM, Theorem 1-3-6] using Theorem 3.14.1for the reader’s convenience.

Theorem 3.14.5 (see Theorem 3.13.5 and [KMM, Theorem 1-3-6]).All weak log-terminal singularities are rational.

Proof. Let (X,∆) be a pair with only weak log-terminal singu-larities. Then we can take a resolution of singularities f : Y → Xwhere

(1) there exists a divisor∑Fj with only normal crossings whose

support is Exc(f) ∪ Supp f−1∗ ∆,

(2) KY = f ∗(KX + ∆) +∑ajFj with the condition that aj > −1

whenever Fj is f -exceptional, and(3) there exists an f -ample Cartier divisor A =

∑bjFj where

bj = 0 if Fj is not f -exceptional.

Note that bj ≤ 0 for every j, equivalently, −A is effective, by thenegativity lemma (see Lemma 2.3.26). We put

J ′ = j |Fj is f -exceptional

and

J ′′ = j |Fj is not f -exceptional.We put

E ′ =∑j∈J ′

ajFj, E = dE ′e

3.15. METHOD OF TWO SPECTRAL SEQUENCES 95

andD =

∑j∈J ′′

(−ajFj) + E − E ′ − δA

for some sufficiently small number δ. Then

L = KY − E, L = 0, and D = 0

satisfy the conditions (a) and (b) in Remark 3.14.4 and (iii) and (iv)of Theorem 3.14.1. Therefore, we obtain

0 = Rif∗OY (L) = Rif∗OYfor every i > 0.

3.15. Method of two spectral sequences

In this section, we give a proof of the following well-known theoremagain (see Theorem 3.13.1).

Theorem 3.15.1. Let (X,D) be a dlt pair. Then X has only ra-tional singularities.

Our proof is a combination of the proofs in [KoMo, Theorem 5.22]and [Ko8, Section 11]. We need no difficult duality theorems.

Let us give a dual form of the Grauert–Riemenschneider vanishingtheorem: Theorem 3.2.7.

Lemma 3.15.2 (see also Lemma 7.1.2). Let f : Y → X be a properbirational morphism from a smooth variety Y to a variety X. Letx ∈ X be a closed point. We put F = f−1(x). Then we have

H iF (Y,OY ) = 0

for every i < n = dimX.

Proof. We take a proper birational morphism g : Z → Y froma smooth variety Z such that f g is projective. We consider thefollowing spectral sequence

Epq2 = Hp

F (Y,Rqg∗OZ)⇒ Hp+qE (Z,OZ),

where E = g−1(F ) = (f g)−1(x). Since Rqg∗OZ = 0 for q > 0 andg∗OZ ' OY , we have Hp

F (Y,OY ) ' HpE(Z,OZ) for every p. Therefore,

we can replace Y with Z and assume that f : Y → X is projective.Without loss of generality, we may assume that X is affine. Then wecompactify X and assume that X and Y are projective. It is wellknown that

H iF (Y,OY ) ' lim

−→m

Exti(OmF ,OY )

96 3. CLASSICAL VANISHING THEOREMS AND SOME APPLICATIONS

(see [Har2, Theorem 2.8]) and that

Hom(Exti(OmF ,OY ),C) ' Hn−i(Y,OmF ⊗ ωY )

by duality on a smooth projective variety Y (see [Har4, Theorem 7.6(a)]). Therefore,

Hom(H iF (Y,OY ),C) ' Hom(lim

−→m

Exti(OmF ,OY ),C)

' lim←−m

Hn−i(Y,OmF ⊗ ωY )

' (Rn−if∗ωY )∧x

by the theorem on formal functions (see [Har4, Theorem 11.1]), where(Rn−if∗ωY )∧x is the completion of Rn−if∗ωY at x ∈ X. On the otherhand, Rn−if∗ωY = 0 for i < n by the Grauert–Riemenschneider van-ishing theorem: Theorem 3.2.7. Thus, H i

F (Y,OY ) = 0 for i < n.

Remark 3.15.3. Lemma 3.15.2 holds true even when Y has rationalsingularities. This is because Rqg∗OZ = 0 for q > 0 and g∗OZ ' OYholds in the proof of Lemma 3.15.2.

Let us go to the proof of Theorem 3.15.1.

Proof of Theorem 3.15.1. Without loss of generality, we mayassume that X is affine. Moreover, by taking general hyperplane sec-tions of X, we may also assume that X has only rational singularitiesoutside a closed point x ∈ X. By the definition of dlt, we can take aresolution f : Y → X such that Exc(f) and Exc(f) ∪ Supp f−1

∗ D areboth simple normal crossing divisors on Y ,

KY + f−1∗ D = f ∗(KX +D) + E

with dEe ≥ 0, and that f is projective. Moreover, we can make fan isomorphism over the generic point of any log canonical center of(X,D) (see Remark 2.3.18). Therefore, by Lemma 3.2.11, we can checkthat Rif∗OY (dEe) = 0 for every i > 0. We note that f∗OY (dEe) ' OXsince dEe is effective and f -exceptional. For every i > 0, by the aboveassumption, Rif∗OY is supported at a point x ∈ X if it ever has a non-empty support at all. We put F = f−1(x). Then we have a spectralsequence

Ei,j2 = H i

x(X,Rjf∗OY (dEe))⇒ H i+j

F (Y,OY (dEe)).

By the above vanishing result, we have

H ix(X,OX) ' H i

F (Y,OY (dEe))

3.15. METHOD OF TWO SPECTRAL SEQUENCES 97

for every i ≥ 0. We obtain a commutative diagram

H iF (Y,OY ) // H i

F (Y,OY (dEe))

H ix(X,OX)

α

OO

H ix(X,OX).

β

OO

We have already checked that β is an isomorphism for every i and thatH iF (Y,OY ) = 0 for i < n (see Lemma 3.15.2). Therefore, H i

x(X,OX) =0 for every i < n = dimX. Thus, X is Cohen–Macaulay. For i = n,we obtain that

α : Hnx (X,OX)→ Hn

F (Y,OY )

is injective. We consider the following spectral sequence

Ei,j2 = H i

x(X,Rjf∗OY )⇒ H i+j

F (Y,OY ).

We note that H ix(X,R

jf∗OY ) = 0 for every i > 0 and j > 0 sinceSuppRjf∗OY ⊂ x for j > 0. On the other hand,

Ei,02 = H i

x(X,OX) = 0

for every i < n. Therefore,

H0x(X,R

jf∗OY ) ' Hjx(X,OX) = 0

for all j ≤ n − 2. Thus, Rjf∗OY = 0 for 1 ≤ j ≤ n − 2. SinceHn−1x (X,OX) = 0, we obtain that

0→ H0x(X,R

n−1f∗OY )→ Hnx (X,OX)

α→ HnF (Y,OY )→ 0

is exact. We have already checked that α is injective. So, we obtainthat H0

x(X,Rn−1f∗OY ) = 0. This means that Rn−1f∗OY = 0. Thus,

we have Rif∗OY = 0 for every i > 0. We complete the proof.

Remark 3.15.4. The method of two spectral sequences was intro-duced in the proof of [KoMo, Theorem 5.22]. In [Ale3], Alexeev usedthis method in order to establish his criterion for Serre’s S3 condition.The method of two spectral sequences of local cohomology groups dis-cussed in this section was first used in [F17, Section 4.3] in order togeneralize Alexeev’s criterion for S3 condition (see Section 7.1). Theproof of Theorem 3.15.1 in this section first appeared in [F17, Subsec-tion 4.2.1].

98 3. CLASSICAL VANISHING THEOREMS AND SOME APPLICATIONS

3.16. Toward new vanishing theorems

In [F28], the following results played crucial roles. For the proofand the details, see [F28].

Proposition 3.16.1 (see, for example, [F28, Proposition 5.1]) can beproved by the theory of mixed Hodge structures. Note that Theorem5.4.1 below is a complete generalization of Proposition 3.16.1.

Proposition 3.16.1 (Fundamental injectivity theorem). Let X bea smooth projective variety and let S + B be a boundary R-divisoron X such that the support of S + B is simple normal crossing andbS+Bc = S. Let L be a Cartier divisor on X and let D be an effectiveCartier divisor whose support is contained in SuppB. Assume thatL ∼R KX + S +B. Then the natural homomorphisms

Hq(X,OX(L))→ Hq(X,OX(L+D))

which are induced by the natural inclusion OX → OX(D) are injectivefor all q.

Proposition 3.16.1 is one of the correct generalizations of Kollar’sinjectivity theorem (see Theorem 3.6.2) from the Hodge theoretic view-point. By Proposition 3.16.1, we can prove Theorem 3.16.2 (see, forexample, [F28, Theorem 6.1]), which is a generalization of Kollar’s in-jectivity theorem. Theorem 5.6.2 below is a generalization of Theorem3.16.2 for simple normal crossing pairs.

Theorem 3.16.2 (Injectivity theorem). Let X be a smooth pro-jective variety and let ∆ be a boundary R-divisor such that Supp ∆ issimple normal crossing. Let L be a Cartier divisor on X and let Dbe an effective Cartier divisor that contains no log canonical centers of(X,∆). Assume the following conditions.

(i) L ∼R KX + ∆ +H,(ii) H is a semi-ample R-divisor, and(iii) tH ∼R D + D′ for some positive real number t, where D′ is

an effective R-Cartier R-divisor whose support contains no logcanonical centers of (X,∆).

Then the homomorphisms

Hq(X,OX(L))→ Hq(X,OX(L+D))

which are induced by the natural inclusion OX → OX(D) are injectivefor all q.

There are no difficulties to prove Theorem 3.16.3 as an applica-tion of Theorem 3.16.2 (see, for example, [F28, Theorem 6.3]). Theo-rem 3.16.3 contains Kollar’s torsion-free theorem and Kollar’s vanishing

3.16. TOWARD NEW VANISHING THEOREMS 99

theorem (see Theorem 3.6.3). We will prove a generalization of Theo-rem 3.16.3 for simple normal crossing pairs (see Theorem 5.6.3 below).Theorem 5.6.3 is a key ingredient of the theory of quasi-log schemesdiscussed in Chapter 6.

Theorem 3.16.3 (Torsion-freeness and vanishing theorem). Let Ybe a smooth variety and let ∆ be a boundary R-divisor such that Supp ∆is simple normal crossing. Let f : Y → X be a projective morphismand let L be a Cartier divisor on Y such that L− (KY + ∆) is f -semi-ample.

(i) Let q be an arbitrary non-negative integer. Then every associ-ated prime of Rqf∗OY (L) is the generic point of the f -imageof some stratum of (Y,∆).

(ii) Let π : X → S be a projective morphism. Assume that

L− (KY + ∆) ∼R f∗H

for some π-ample R-divisor H on X. Then

Rpπ∗Rqf∗OY (L) = 0

for every p > 0 and q ≥ 0.

As an easy consequence of Theorem 3.16.3, we obtain Theorem3.16.4 in [F28].

Theorem 3.16.4 (see [F28, Theorem 8.1]). Let X be a normalvariety and let ∆ be an effective R-divisor on X such that KX + ∆ isR-Cartier. Let D be a Cartier divisor on X. Assume that D−(KX+∆)is π-ample, where π : X → S is a projective morphism onto a varietyS. Let Ci be any set of log canonical centers of the pair (X,∆). Weput W =

∪Ci with the reduced scheme structure. Assume that W is

disjoint from the non-lc locus Nlc(X,∆) of (X,∆). Then we have

Riπ∗(J ⊗OX(D)) = 0

for every i > 0, where J = IW · JNLC(X,∆) ⊂ OX and IW is thedefining ideal sheaf of W on X. Therefore, the restriction map

π∗OX(D)→ π∗OW (D)⊕ π∗ONlc(X,∆)(D)

is surjective andRiπ∗OW (D) = 0

for every i > 0. In particular, the restriction maps

π∗OX(D)→ π∗OW (D)

andπ∗OX(D)→ π∗ONlc(X,∆)(D)

100 3. CLASSICAL VANISHING THEOREMS AND SOME APPLICATIONS

are surjective.

In [F28], Theorem 3.16.5 (see [F28, Theorem 11.1]) plays crucialroles for the proof of the non-vanishing theorem in [F28, Theorem12.2].

Theorem 3.16.5 (Vanishing theorem for minimal log canonical cen-ters). Let X be a normal variety and let ∆ be an effective R-divisor onX such that KX + ∆ is R-Cartier. Let W be a minimal log canonicalcenter of (X,∆) such that W is disjoint from the non-lc locus Nlc(X,∆)of (X,∆). Let π : X → S be a projective morphism onto a variety S.Let D be a Cartier divisor on W such that D−(KX+∆)|W is π-ample.Then Riπ∗OW (D) = 0 for every i > 0.

In [F28], we proved Theorem 3.16.5 by using dlt blow-ups (see The-orem 4.4.21), which depend on the recent developments of the minimalmodel program, and Theorem 3.16.3. Therefore, Theorem 3.16.5 ismuch harder than Theorem 3.16.4. Note that Theorem 3.16.4 andTheorem 3.16.5 are special cases of Theorem 6.3.4 below. The proof ofTheorem 6.3.4 in Chapter 6 does not need the minimal model programbut uses the theory of mixed Hodge structures for reducible varieties.

In [F28], we obtained the fundamental theorems, that is, variousKodaira type vanishing theorem, the cone and contraction theorem,and so on, for normal pairs by using Theorem 3.16.3 and Theorem3.16.5 (see Section 4.5). Our formulation in [F28] is different fromthe traditional X-method and is similar to the theory of (algebraic)multiplier ideal sheaves based on the Nadel vanishing theorem (see, forexample, [La2, Part Three]). In [F28], we need no vanishing theoremsfor reducible varieties.

Remark 3.16.6. Theorem 3.16.2 (resp. Theorem 3.16.3) is a specialcase of [Am1, Theorem 3.1] (resp. [Am1, Theorem 3.2]). The proofof [Am1, Theorem 3.1] contains several difficulties (see, for example,Example 5.1.4). Moreover, Ambro’s original proof of [Am1, Theorem3.2 (ii)] used [Am1, Theorem 3.2 (i)] for embedded normal crossingpairs even when Y is smooth in [Am1, Theorem 3.2 (ii)]. On the otherhand, the proof of Theorem 3.16.3 in [F28] does not need reduciblevarieties.

In Chapter 5 and Chapter 6, we will prove more general resultsthan the theorems in this section. Note that [KMM, Theorem 1-2-5and Remark 1-2-6] will be generalized as follows.

Theorem 3.16.7 (Theorem 5.7.6). Let (X,∆) be a log canonicalpair such that ∆ is a boundary R-divisor and let L be a Q-Cartier Weil

3.16. TOWARD NEW VANISHING THEOREMS 101

divisor on X. Assume that L − (KX + ∆) is nef and log big over Vwith respect to (X,∆), where π : X → V is a proper morphism. ThenRqπ∗OX(L) = 0 for every q > 0.

The proof of Theorem 3.16.7 (see Theorem 5.7.6) needs the vanish-ing theorem for reducible varieties. Therefore, it is much harder thanthe arguments in this chapter.

We strongly recommend the reader to see [F28, Section 3], where wediscussed the conceptual difference between the traditional argumentsbased on the Kawamata–Viehweg–Nadel vanishing theorem and ournew approach depending on the theory of mixed Hodge structures.It will help the reader to understand the results and the frameworkdiscussed in Chapter 5 and Chapter 6.

CHAPTER 4

Minimal model program

In this chapter, we discuss the minimal model program. Althoughwe explain the recent developments of the minimal model programmainly due to Birkar–Cascini–Hacon–McKernan in Section 4.4, we donot discuss the proof of the main results of [BCHM]. For the details of[BCHM], see [BCHM], [HaKo, Part II], [HaMc1], [HaMc2], and soon. The papers [Dr], [F25], and [Ka4] are survey articles on [BCHM].For slightly different approaches, see [BirPa], [CoLa], [CaL], [P], andso on. In this book, we mainly discuss the topics of the minimal modelprogram which are not directly related to [BCHM].

In Sections 4.1, 4.2, and 4.3, we quickly review the basic resultson the minimal model program, X-method, and so on. Section 4.4is devoted to the explanation on [BCHM] and some related resultsand examples. In Section 4.5, we discuss the fundamental theoremsfor normal pairs (see [F28]) and various examples of the Kleiman–Mori cone. The results in Section 4.5 are sufficient for the minimalmodel program for log canonical pairs. In Section 4.6 and Section 4.7,we prove that Shokurov polytope is a polytope. In Section 4.8, wediscuss the minimal model program for log canonical pairs and variousconjectures. In Section 4.9, we explain the minimal model program for(not necessarily Q-factorial) log canonical pairs. It is the most generalminimal model program in the usual sense. In Section 4.10, we reviewthe minimal model theory for singular surfaces following [F29]. InSection 4.11, we quickly explain the author’s recent result on semi logcanonical pairs without proof.

4.1. Fundamental theorems for klt pairs

In this section, we assume that X is a projective irreducible va-riety and ∆ is an effective Q-divisor for simplicity. Let us recall thefundamental theorems for klt pairs. For the details, see, for exam-ple, [KoMo, Chapter 3]. A starting point is the following vanishingtheorem (see Theorem 3.1.7).

Theorem 4.1.1 (Kawamata–Viehweg vanishing theorem). Let Xbe a smooth projective variety and let D be a Q-divisor such that

103

104 4. MINIMAL MODEL PROGRAM

SuppD is a simple normal crossing divisor on X. Assume that D isample. Then

H i(X,OX(KX + dDe)) = 0

for every i > 0.

The next theorem is Shokurov’s non-vanishing theorem (see [Sh1]).

Theorem 4.1.2 (Non-vanishing theorem). Let X be a projectivevariety, let D be a nef Cartier divisor, and let G be a Q-divisor. Sup-pose

(i) aD +G−KX is an ample Q-divisor for some a > 0, and(ii) (X,−G) is sub klt.

Then there is a positive integer m0 such that

H0(X,OX(mD + dGe)) 6= 0

for every m ≥ m0.

It plays important roles in the proof of the basepoint-free and ra-tionality theorems below.

Theorem 4.1.3 (Basepoint-free theorem). Let (X,∆) be a projec-tive klt pair. Let D be a nef Cartier divisor such that aD − (KX + ∆)is ample for some a > 0. Then there is a positive integer b0 such that|bD| has no base points for every b ≥ b0.

Theorem 4.1.4 (Rationality theorem). Let (X,∆) be a projectiveklt pair such that KX + ∆ is not nef. Let a > 0 be an integer such thata(KX + ∆) is Cartier. Let H be an ample Cartier divisor. We define

r = max t ∈ R |H + t(KX + ∆) is nef .Then r is a rational number of the form u/v, where u and v are integerswith

0 < v ≤ a(dimX + 1).

The final theorem is the cone and contraction theorem. It easily fol-lows from the basepoint-free and rationality theorems: Theorems 4.1.3and 4.1.4.

Theorem 4.1.5 (Cone and contraction theorem). Let (X,∆) be aprojective klt pair. Then we have the following properties.

(i) There are (countably many possibly singular) rational curvesCj ⊂ X such that

NE(X) = NE(X)(KX+∆)≥0 +∑

R≥0[Cj].

4.2. X-METHOD 105

(ii) Let R ⊂ NE(X) be a (KX + ∆)-negative extremal ray. Thenthere is a unique morphism ϕR : X → Z to a projective varietyZ such that (ϕR)∗OX ' OZ and an irreducible curve C ⊂ Xis mapped to a point by ϕR if and only if [C] ∈ R.

We note that the cone and contraction theorem can be proved fordlt pairs in the relative setting (see, for example, [KMM]). We omit ithere because we will give a complete generalization of the cone and con-traction theorem for quasi-log schemes in Chapter 6. See also Theorem4.5.2 below.

The main purpose of this book is to establish the cone and contrac-tion theorem for quasi-log schemes (see Chapter 6). Note that a logcanonical pair has a natural quasi-log structure which is compatiblewith the original log canonical structure.

4.2. X-method

In this section, we give a proof of the basepoint-free theorem forklt pairs (see Theorem 4.1.3) by assuming the non-vanishing theorem(see Theorem 4.1.2). The following proof is taken almost verbatimfrom [KoMo, 3.2 Basepoint-free Theorem]. This type of argument isusually called X-method. It has various applications in many differentcontexts.

Proof of the basepoint-free theorem: Theorem 4.1.3. Weprove the basepoint-free theorem.

Step 1. In this step, we establish that |mD| 6= ∅ for every m 0.We can construct a resolution of singularities f : Y → X such that

(i) KY = f ∗(KX + ∆) +∑ajFj with all aj > −1,

(ii) f∗(aD− (KX + ∆))−∑pjFj is ample for some a > 0 and for

suitable 0 < pj 1, and(iii)

∑Fj(⊃ Exc(f) ∪ Supp f−1

∗ ∆) is a simple normal crossing di-visor on Y .

We note that the Fj is not necessarily f -exceptional. On Y , we write

f∗(aD − (KX + ∆))−∑

pjFj

= af ∗D +∑

(aj − pj)Fj − (f ∗(KX + ∆) +∑

ajFj)

= af ∗D +G−KY ,

where G =∑

(aj − pj)Fj. By assumption, dGe is an effective f -exceptional divisor, af ∗D +G−KY is ample, and

H0(Y,OY (mf ∗D + dGe)) ' H0(X,OX(mD)).

106 4. MINIMAL MODEL PROGRAM

We can now apply the non-vanishing theorem (see Theorem 4.1.2) toget that H0(X,OX(mD)) 6= 0 for every m 0.

Step 2. For a positive integer s, let B(s) denote the reduced baselocus of |sD|. Clearly, we have B(su) ⊂ B(sv) for every positive inte-gers u > v. The noetherian induction implies that the sequence B(su)stabilizes, and we call the limit Bs. So either Bs is non-empty for somes or Bs and Bs′ are empty for two relatively prime integers s and s′. Inthe latter case, take u and v such that B(su) and B(s′v) are empty, anduse the fact that every sufficiently large integer is a linear combinationof su and s′v with non-negative coefficients to conclude that |mD| isbasepoint-free for every m 0. So, we must show that the assumptionthat some Bs is non-empty leads to a contradiction. We let m = su

such that Bs = B(m) and assume that this set is non-empty.Starting with the linear system obtained from Step 1, we can blow

up further to obtain a new f : Y → X for which the conditions of Step1 hold, and, for some m > 0,

f∗|mD| = |L| (moving part) +∑

rjFj (fixed part)

such that |L| is basepoint-free. Therefore,∪f(Fj)|rj > 0 is the base

locus of |mD|. Note that f−1 Bs |mD| = Bs |mf ∗D|. We obtain thedesired contradiction by finding some Fj with rj > 0 such that, forevery b 0, Fj is not contained in the base locus of |bf ∗D|.

Step 3. For an integer b > 0 and a rational number c > 0 suchthat b ≥ cm+ a, we define divisors:

N(b, c) = bf ∗D −KY +∑

(−crj + aj − pj)Fj= (b− cm− a)f ∗D (nef)

+c(mf ∗D −∑

rjFj) (basepoint-free)

+f∗(aD − (KX + ∆))−∑

pjFj (ample).

Thus, N(b, c) is ample for b ≥ cm + a. If that is the case, then, byTheorem 4.1.1, H1(Y,OY (dN(b, c)e+KY )) = 0, and

dN(b, c)e = bf ∗D +∑d−crj + aj − pjeFj −KY .

Step 4. c and pj can be chosen so that∑(−crj + aj − pj)Fj = A− F

4.3. MMP FOR Q-FACTORIAL DLT PAIRS 107

for some F = Fj0 , where dAe is effective and A does not have F as acomponent. In fact, we choose c > 0 so that

minj

(−crj + aj − pj) = −1.

If this last condition does not single out a unique j, we wiggle the pjslightly to achieve the desired uniqueness. This j satisfies rj > 0 anddN(b, c)e+KY = bf ∗D + dAe − F . Now Step 3 implies that

H0(Y,OY (bf ∗D + dAe))→ H0(F,OF (bf ∗D + dAe))is surjective for b ≥ cm+ a. If Fj appears in dAe, then aj > 0, so Fj isf -exceptional. Thus, dAe is f -exceptional.

Step 5. Notice that

N(b, c)|F = (bf ∗D + A− F −KY )|F = (bf ∗D + A)|F −KF .

So we can apply the non-vanishing theorem (see Theorem 4.1.2) on Fto get

H0(F,OF (bf ∗D + dAe)) 6= 0.

Thus, H0(Y,OY (bf ∗D+ dAe)) has a section not vanishing on F . SincedAe is f -exceptional and effective,

H0(Y,OY (bf ∗D + dAe)) ' H0(X,OX(bD)).

Therefore, f(F ) is not contained in the base locus of |bD| for everyb 0.

This completes the proof of the basepoint-free theorem. The X-method is very powerful and very useful for klt pairs. Un-

fortunately, it can not be applied for log canonical pairs. So we needthe framework discussed in [F28] or the theory of quasi-log schemes(see Chapter 6) in order to treat log canonical pairs. For the details ofX-method, see [KMM] and [KoMo].

We note that the X-method, the technique which was used for theproofs of Theorems 4.1.2, 4.1.3, 4.1.4, and 4.1.5, was developed byseveral authors. The main contributions are [Ka2], [Ko1], [R1], and[Sh1].

4.3. MMP for Q-factorial dlt pairs

In this section, we quickly explain the minimal model program forQ-factorial dlt pairs. First, let us recall the definition of the (log)minimal models. Definition 4.3.1 is a traditional definition of minimalmodels. For slightly different other definitions of minimal models, seeDefinition 4.4.4 and Definition 4.8.5.

108 4. MINIMAL MODEL PROGRAM

Definition 4.3.1 ((Log) minimal model). Let (X,∆) be a logcanonical pair and let f : X → S be a proper morphism. A pair(X ′,∆′) sitting in a diagram

Xφ //_______

f @@@

@@@@

X ′

f ′~~

S

is called a (log) minimal model of (X,∆) over S if

(i) f ′ is proper,(ii) φ−1 has no exceptional divisors,(iii) ∆′ = φ∗∆,(iv) KX′ + ∆′ is f ′-nef, and(v) a(E,X,∆) < a(E,X ′,∆′) for every φ-exceptional divisor E ⊂

X.

Furthermore, if KX′ + ∆′ is f ′-semi-ample, then (X ′,∆′) is called agood minimal model of (X,∆) over S.

We note the following easy lemma.

Lemma 4.3.2. Let (X,∆) be a log canonical pair and let f : X → Sbe a proper morphism. Let (X ′,∆′) be a minimal model of (X,∆) overS. Then a(E,X,∆) ≤ a(E,X ′,∆′) for every divisor E over X.

Proof. We take any common resolution

Wq

!!BBB

BBBB

Bp

~~

X X ′

of X and X ′. Then we can write

KW = p∗(KX + ∆) + F

and

KW = q∗(KX′ + ∆′) +G.

It is sufficient to prove G ≥ F . Note that

p∗(KX + ∆) = q∗(KX′ + ∆′) +G− F.

Then −(G − F ) is p-nef since KX′ + ∆′ is nef over S. Note thatp∗(G− F ) is effective by (v). Therefore, by the negativity lemma (seeLemma 2.3.26), G− F is effective.

4.3. MMP FOR Q-FACTORIAL DLT PAIRS 109

Next, we recall the flip theorem for dlt pairs in [BCHM] and[HaMc1] (see also [HaMc2]). We need the notion of small morphismsto treat flips.

Definition 4.3.3 (Small morphism). Let f : X → Y be a properbirational morphism between normal varieties. If Exc(f) has codimen-sion ≥ 2, then f is called small.

Theorem 4.3.4 ((Log) flip for dlt pairs). Let ϕ : (X,∆) → W bean extremal flipping contraction, that is,

(i) (X,∆) is dlt,(ii) ϕ is small projective and ϕ has connected fibers,(iii) −(KX + ∆) is ϕ-ample,(iv) ρ(X/W ) = 1, and(v) X is Q-factorial.

Then we have the following diagram:

Xφ //_______

ϕ AAA

AAAA

A X+

ϕ+zzzz

zzzz

W

(1) X+ is a normal variety,(2) ϕ+ : X+ → W is small projective, and(3) KX+ + ∆+ is ϕ+-ample, where ∆+ is the strict transform of

∆.

We call ϕ+ : (X+,∆+)→ W a (KX+∆)-flip of ϕ. In this situation, wecan check that (X+,∆+) is a Q-factorial dlt pair with ρ(X+/W ) = 1(see, for example, Lemma 4.8.13 and Proposition 4.8.16 below).

Let us explain the relative minimal model program (MMP, forshort) for Q-factorial dlt pairs.

4.3.5 (MMP for Q-factorial dlt pairs). We start with a pair (X,∆) =(X0,∆0). Let f0 : X0 → S be a projective morphism. The aim is toset up a recursive procedure which creates intermediate pairs (Xi,∆i)and projective morphisms fi : Xi → S. After some steps, it shouldstop with a final pair (X ′,∆′) and f ′ : X ′ → S.

Step 0 (Initial datum). Assume that we have already constructed(Xi,∆i) and fi : Xi → S with the following properties:

(i) Xi is Q-factorial,(ii) (Xi,∆i) is dlt, and(iii) fi is projective.

110 4. MINIMAL MODEL PROGRAM

Step 1 (Preparation). If KXi+ ∆i is fi-nef, then we go directly

to Step 3 (ii). If KXi+ ∆i is not fi-nef, then we have established the

following two results:

(i) (Cone theorem) We have the following equality.

NE(Xi/S) = NE(Xi/S)(KXi+∆i)≥0 +

∑R≥0[Ci].

(ii) (Contraction theorem) Any (KXi+ ∆i)-negative extremal ray

Ri ⊂ NE(Xi/S) can be contracted. Let ϕRi: Xi → Yi de-

note the corresponding contraction. It sits in a commutativediagram.

Xi

ϕRi //

fi @@@

@@@@

@Yi

gi

S

Step 2 (Birational transformations). If ϕRi: Xi → Yi is birational,

then we produce a new pair (Xi+1,∆i+1) as follows.

(i) (Divisorial contraction). If ϕRiis a divisorial contraction, that

is, ϕRicontracts a divisor, then we set Xi+1 = Yi, fi+1 = gi,

and ∆i+1 = (ϕRi)∗∆i.

(ii) (Flipping contraction). If ϕRiis a flipping contraction, that

is, ϕRiis small, then we set (Xi+1,∆i+1) = (X+

i ,∆+i ), where

(X+i ,∆

+i ) is the flip of ϕRi

(Xi,∆i)

ϕRi ##GGGGGGGGG

//_______ (X+i ,∆

+i )

ϕ+Rizzuuuu

uuuuuu

Yi

and fi+1 = gi ϕ+Ri

(see Theorem 4.3.4).

In both cases, we can prove that Xi+1 is Q-factorial, fi+1 is projec-tive and (Xi+1,∆i+1) is dlt (see, for example, Lemma 4.8.13, Proposi-tion 4.8.14, and Proposition 4.8.16). Then we go back to Step 0 with(Xi+1,∆i+1) and start anew.

Step 3 (Final outcome). We expect that eventually the procedurestops, and we get one of the following two possibilities:

(i) (Mori fiber space). If ϕRiis a Fano contraction, that is, dimYi <

dimXi, then we set (X ′,∆′) = (Xi,∆i) and f ′ = fi. In thiscase, we usually call f ′ : (X ′,∆′) → Yi a Mori fiber space of(X,∆) over S.

4.3. MMP FOR Q-FACTORIAL DLT PAIRS 111

(ii) (Minimal model). If KXi+ ∆i is fi-nef, then we again set

(X ′,∆′) = (Xi,∆i) and f ′ = fi. We can easily check that(X ′,∆′) is a minimal model of (X,∆) over S in the sense ofDefinition 4.3.1.

By the results in [BCHM] and [HaMc1] (see also [HaMc2]), allwe have to do is to prove that there are no infinite sequence of flips inthe above process.

Conjecture 4.3.6 (Flip conjecture II). A sequence of (log) flips

(X0,∆0) 99K (X1,∆1) 99K · · · 99K (Xi,∆i) 99K · · ·

terminates after finitely many steps. Namely there does not exist aninfinite sequence of (log) flips.

Remark 4.3.7. In Conjecture 4.3.6, each flip

(Xi,∆i) 99K (Xi+1,∆i+1)

is a flip as in Theorem 4.3.4.

Lemma 4.3.8. We assume that Conjecture 4.3.6 holds in the fol-lowing two cases:

(i) (X0,∆0) is klt with dimX0 = n, and(ii) (X0,∆0) is dlt with dimX0 ≤ n− 1.

Then Conjecture 4.3.6 holds for n-dimensional dlt pair (X0,∆0). There-fore, by induction on the dimension, it is sufficient to prove Conjecture4.3.6 under the extra assumption that (X0,∆0) is klt.

Proof. Let

(X0,∆0) 99K (X1,∆1) 99K · · · 99K (Xi,∆i) 99K · · ·

be a sequence of flips as in Conjecture 4.3.6 with dimX0 = n. By thecase (ii), the special termination theorem holds in dimension n (see, forexample, [F13, Theorem 4.2.1]). Therefore, after finitely many steps,the flipping locus (and thus the flipped locus) is disjoint from b∆ic.Thus, we may assume that b∆ic = 0 by replacing ∆i with ∆i. Inthis case, the above sequence terminates by the case (i).

Conjecture 4.3.6 was completely solved in dimension ≤ 3 (see, forexample, [Koetal, Chapter 6] and [Sh3, 5.1.3]). Conjecture 4.3.6 isstill open even when dimX0 = 4. For the details of Conjecture 4.3.6 indimension 4, see [KMM, Theorem 5-1-15], [F5], [F7], [F8], [AHK],and [Bir1].

112 4. MINIMAL MODEL PROGRAM

4.4. BCHM and some related results

In this section, we quickly review the main results of [HaMc1],[HaMc2], and [BCHM] for the reader’s convenience. We also discusssome related results. We closely follow the presentation of [HaKo,5.D].

Roughly speaking, [BCHM] established:

Theorem 4.4.1 (Minimal model program). Let π : X → S be aprojective morphism of normal quasi-projective varieties and let (X,∆)be a Q-factorial klt pair such that ∆ is π-big. Then there exists a finitesequence of flips and divisorial contractions for the (KX + ∆)-minimalmodel program over S:

X = X0 99K X1 99K · · · 99K XN

such that either KXN+ ∆N is nef over S or there exists a morphism

XN → Z which is a (KXN+ ∆N)-Mori fiber space over S.

From now on, let us explain the results in [BCHM] more details.

Definition 4.4.2 (Pl-flipping contraction). Let (X,∆) be a pltpair with S = b∆c. A pl-flipping contraction is a flipping contractionϕ : X → W , that is, ϕ is small, ϕ∗OX ' OW , ρ(X/W ) = 1, and−(KX + ∆) is ϕ-ample, such that ∆ is a Q-divisor, S is irreducible,and −S is ϕ-ample.

For the definition of pl-flipping contractions, see also [F13, Def-inition 4.3.1 and Caution 4.3.2]. Note that the notion of pl-flippingcontractions and pl-flips is due to Shokurov (see [Sh2]).

Theorem 4.4.3 (Existence of pl-flips). Let (X,∆) be a plt pair andlet ϕ : X → W be a pl-flipping contraction. Then the flip

ϕ+ : X+ → W

of ϕ exists.

In this section, we adopt the following definition of minimal models,which is slightly different from Definition 4.3.1.

Definition 4.4.4 (Minimal models). Let (X,∆) be a dlt pair andlet π : X → S be a projective morphism onto a variety S. Let φ :X 99K Y be a rational map over S such that

(i) φ−1 contracts no divisors,(ii) Y is Q-factorial,(iii) KY + φ∗∆ is nef over S, and(iv) a(E,X,∆) < a(E, Y, φ∗∆) for every φ-exceptional divisor E

on X.

4.4. BCHM AND SOME RELATED RESULTS 113

Then (Y, φ∗∆) is called a minimal model of (X,∆) over S. Further-more, if KY +φ∗∆ is semi-ample over S, then (Y, φ∗∆) is called a goodminimal model of (X,∆) over S.

It is obvious that a minimal model in the sense of Definition 4.4.4is a minimal model in the sense of Definition 4.3.1.

Theorem 4.4.5 (Existence of minimal models). Let π : X → S bea projective morphism of normal quasi-projective varieties. Let (X,∆)be a klt pair and let D be an effective R-divisor on X such that ∆ is π-big and KX + ∆ ∼R,π D. Then there exists a minimal model of (X,∆)over S.

Theorem 4.4.6 (Non-vanishing theorem). Let π : X → S be aprojective morphism of normal quasi-projective varieties. Let (X,∆)be a klt pair such that ∆ is π-big. If KX +∆ is pseudo-effective over S,then there exists an effective R-divisor D on X such that KX +∆ ∼R,πD.

Definition 4.4.7. On a normal variety X, the group of Weil divi-sors with rational coefficients Weil(X)Q, or with real coefficients Weil(X)R,forms a vector space, with a canonical basis given by the prime divi-sors. Let D be an R-divisor on X. Then ||D|| denotes the sup normwith respect to this basis.

Theorem 4.4.8 (Finiteness of marked minimal models). Let π :X → S be a projective morphism of normal quasi-projective varieties.Let C ⊂ Weil(X)R be a rational polytope such that for every KX +∆ ∈ C, ∆ is π-big, and (X,∆) is klt. Then there exist finitely manybirational maps φi : X 99K Yi over S with 1 ≤ i ≤ k such that ifKX + ∆ ∈ C and KX + ∆ is pseudo-effective over S, then

(i) There exists an index 1 ≤ j ≤ k such that φj : X 99K Yj is aminimal model of (X,∆) over S.

(ii) If φ : X 99K Y is a minimal model of (X,∆) over S, thenthere exists an index 1 ≤ j ≤ k such that the rational mapφj φ−1 : Y 99K Yj is an isomorphism.

We need the notion of stable base locus and stable augmented baselocus.

Definition 4.4.9 (Stable base locus and stable augmented baselocus). Let π : X → S be a morphism from a normal variety X onto avariety S. The real linear system over S associated to an R-divisor Don X is

|D/S|R = D′ ≥ 0 |D′ ∼R,π D.

114 4. MINIMAL MODEL PROGRAM

We can define

|D/S|Q = D′ ≥ 0 |D′ ∼Q,π Dsimilarly. The stable base locus of D over S is the Zariski closed subset

B(D/S) =∩

D′∈|D/S|R

SuppD′.

If |D/S|R = ∅, then we put B(D/S) = X. When D is Q-Cartier,B(D/S) is the usual stable base locus (see [BCHM, Lemma 3.5.3]).When S is affine, we sometimes simply use B(D) to denote B(D/S).

The stable augmented base locus of D over S is the Zariski closedset

B+(D/S) = B((D − εA)/S)

for any π-ample R-divisor A and any sufficiently small rational numberε > 0.

Let Λ be a non-empty linear system on X. Then the fixed divisorFix Λ is the largest effective divisor F on X such that D ≥ F for allD ∈ Λ.

Theorem 4.4.10 (Zariski decomposition). Let π : X → S be aprojective morphisim to a normal affine variety S. Let (X,∆) be a kltpair where KX + ∆ is pseudo-effective over S, ∆ = A + B, A is anample effective Q-divisor, and B is an effective R-divisor. Then wehave the following properties.

(i) (X,∆) has a minimal model φ : X 99K Y over S. In particu-lar, if KX + ∆ is Q-Cartier, then the log canonical ring

R(X,KX + ∆) =⊕m≥0

H0(X,OX(bm(KX + ∆)c)

is finitely generated.(ii) Let V ⊂ Weil(X)R be a finite dimensional affine subspace of

Weil(X)R containing ∆ which is defined over Q. Then thereexists a constant δ > 0 such that if P is a prime divisor con-tained in B(KX + ∆), then P is contained in B(KX + ∆′) forany R-divisor ∆′ ∈ V with ||∆−∆|| ≤ δ.

(iii) Let W ⊂ Weil(X)R be the smallest affine subspace containing∆ which is defined over Q. Then there exist a real number η >0 and a positive integer r such that if ∆′ ∈ W , ||∆ −∆′|| ≤ ηand k is a positive integer such that k(KX + ∆′)/r is Cartier,then |k(KX+∆′)| 6= ∅ and every component of Fix |k(KX+∆′)|is a component of B(KX + ∆).

Let us explain the minimal model program with scaling.

4.4. BCHM AND SOME RELATED RESULTS 115

4.4.11 (Minimal model program with scaling). Let (X,∆ + C) bea log canonical pair and let π : X → S be a projective morphism ontoa variety S such that KX + ∆ +C is π-nef, ∆ is an effective R-divisor,and C is an effective R-Cartier R-divisor on X. We put

(X0,∆0 + C0) = (X,∆ + C).

Assume that KX0 + ∆0 is nef over S or there exists a (KX0 + ∆0)-negative extremal ray R0 over S such that (KX0 + ∆0 + λ0C0) ·R0 = 0where

λ0 = inft ≥ 0 |KX0 + ∆0 + tC0 is nef over S.If KX0 + ∆0 is nef over S or if R0 defines a Mori fiber space structureover S, then we stop. Otherwise, we assume that R0 gives a divisorialcontraction X0 → X1 over S or a flip X0 99K X1 over S. We canconsider (X1,∆1 + λ0C1) where ∆1 + λ0C1 is the strict transform of∆0 + λ0C0. Assume that KX1 + ∆1 is nef over S or there exists a(KX1 +∆1)-negative extremal ray R1 such that (KX1 +∆1+λ1C1)·R1 =0 where

λ1 = inft ≥ 0 |KX1 + ∆1 + tC1 is nef over S.By repeating this process, we obtain a sequence of positive real numbersλi and a special kind of the minimal model program over S:

(X0,∆0) 99K (X1,∆1) 99K · · · 99K (Xi,∆i) 99K · · · ,which is called the minimal model program over S on KX + ∆ withscaling of C. We note that λi ≥ λi+1 for every i.

In [KoMo, Section 7.4], it was called a minimal model programover S guided with C.

Theorem 4.4.12 (Termination of flips with scaling). We use thesame notation as in 4.4.11. We assume that (X,∆+C) is a Q-factorialklt pair, S is quasi-projective, and ∆ is π-big. Then we can run theminimal model program with respect to KX + ∆ over S with scaling ofC. Moreover, any sequence of flips and divisorial contractions for the(KX + ∆)-minimal model program over S with scaling of C is finite.

Remark 4.4.13 follows from the argument in [BCHM, Remark3.10.9].

Remark 4.4.13. Let (X,∆) be a Q-factorial dlt pair and let π :X → S be a projective morphism between quasi-projective varieties.Let C be an R-Cartier R-divisor on X such that (X,∆ + C) is logcanonical, B+(C/S) contains no log canonical centers of (X,∆), andKX + ∆ + C is nef over S. Then we can run the minimal modelprogram with respect to KX + ∆ over S with scaling of C. Note that

116 4. MINIMAL MODEL PROGRAM

the termination of this minimal model program is still an open problem.However, it is useful for some applications.

4.4.14 (Finite generation of log canonical rings). By combining[FM, Theorem 5.2] with Theorem 4.4.5, we have:

Theorem 4.4.15. Let (X,∆) be a projective klt pair such that ∆ isa Q-divisor on X. Then the log canonical ring

R(X,KX + ∆) =⊕m≥0

H0(X,OX(bm(KX + ∆)c))

is a finitely generated C-algebra.

As a corollary of Theorem 4.4.15, we obtain:

Corollary 4.4.16. Let X be a smooth projective variety. Thenthe canonical ring

R(X) =⊕m≥0

H0(X,OX(mKX))

is a finitely generated C-algebra.

In [F38], the author obtained the following generalizations of The-orem 4.4.15 and Corollary 4.4.16.

Theorem 4.4.17. Let X be a complex analytic variety in Fujiki’sclass C. Let (X,∆) be a klt pair such that ∆ is a Q-divisor on X. Thenthe log canonical ring

R(X,KX + ∆) =⊕m≥0

H0(X,OX(bm(KX + ∆)c))

is a finitely generated C-algebra.

As a special case of Theorem 4.4.17, we obtain:

Corollary 4.4.18 ([F38, Theorem 5.1]). Let X be a compactKahler manifold, or more generally, let X be a complex manifold inFujiki’s class C. Then the canonical ring

R(X) =⊕m≥0

H0(X,ω⊗mX )

is a finitely generated C-algebra.

Remark 4.4.19 ([F38, Corollary 5.2]). In [W], Wilson constructeda compact complex manifold which is not Kahler whose canonical ringis not a finitely generated C-algebra. For the details, see [F38, Section6].

4.4. BCHM AND SOME RELATED RESULTS 117

4.4.20 (Dlt blow-ups). Let us recall a very important application ofthe minimal model program with scaling. Theorem 4.4.21 is originallydue to Hacon.

Theorem 4.4.21 (Dlt blow-ups). Let X be a normal quasi-projectivevariety and let ∆ be a boundary R-divisor on X such that KX + ∆ isR-Cartier. In this case, we can construct a projective birational mor-phism f : Y → X from a normal quasi-projective variety Y with thefollowing properties.

(i) Y is Q-factorial.(ii) a(E,X,∆) ≤ −1 for every f -exceptional divisor E on Y .(iii) We put

∆Y = f−1∗ ∆ +

∑E:f-exceptional

E.

Then (Y,∆Y ) is dlt and

KY + ∆Y = f∗(KX + ∆) +∑

a(E,X,∆)<−1

(a(E,X,∆) + 1)E.

In particular, if (X,∆) is log canonical, then

KY + ∆Y = f ∗(KX + ∆).

Moreover, if (X,∆) is dlt, then we can make f small, that is,f is an isomorphism in codimension one.

We closely follow the argument in [F26]. For the proof of Theorem4.4.21, see also [F28, Section 10].

Proof. Let g : Z → X be a resolution such that Exc(g)∪Supp g−1∗ ∆

is a simple normal crossing divisor on X and g is projective. We write

KZ + ∆Z = g∗(KX + ∆) + F

where

∆Z = g−1∗ ∆ +

∑E: g-exceptional

E.

Let C be a g-ample effective Q-divisor on Z such that (Z,∆Z + C) isdlt and that KZ+∆Z+C is g-nef. We run the minimal model programwith respect to KZ +∆Z over X with scaling of C (see Remark 4.4.13).We obtain a sequence of divisorial contractions and flips

(Z,∆Z) = (Z0,∆Z0) 99K (Z1,∆Z1) 99K · · · 99K (Zk,∆Zk) 99K · · ·

over X. We note that

λi = inft ∈ R |KZi+ ∆Zi

+ tCi is nef over X,

118 4. MINIMAL MODEL PROGRAM

where Ci (resp. ∆Zi) is the pushforward of C (resp. ∆Z) on Zi for every

i. By definition, 0 ≤ λi ≤ 1, λi ∈ R for every i and

λ0 ≥ λ1 ≥ · · · ≥ λk ≥ · · · .Let Fi be the pushforward of F on Zi for every i. It is sufficient toprove:

Claim. There is i0 such that −Fi0 is effective.

Proof of Claim. If we prove that the above minimal model pro-gram terminates after finitely many steps, then there is i0 such thatFi0 is nef over X. Since Fi0 is exceptional over X, −Fi0 is effective bythe negativity lemma (see Lemma 2.3.26). Therefore, we may assumethat the above minimal model program does not terminate. We put

λ = limi→∞

λi.

Case 1 (λ > 0). In this case, we can see that the above minimalmodel program is a minimal model program with respect to (KZ +∆Z +12λC) over X with scaling of (1− 1

2λ)C. By assumption, we can write

∆Z +1

2λC ∼R,π B

such that (Z,B) and (Z,B + (1 − 12λ)C) are klt. Therefore, it is a

minimal model program with respect to KZ +B over X with scaling of(1− 1

2λ)C. This contradicts Theorem 4.4.12.

Case 2 (λ = 0). After finitely many steps, every step of the aboveminimal model program is flip. Therefore, without loss of generality,we may assume that all the steps are flips. Let Gi be a relative ampleQ-divisor on Zi such that GiZ → 0 in N1(Z/X) for i→∞ where GiZ

is the strict transform of Gi on Z. We note that

KZi+ ∆Zi

+ λiCi +Gi

is ample over X for every i. Therefore, the strict transform

KZ + ∆Z + λiC +GiZ

is movable on Z for every i. Thus KZ + ∆Z is a limit of movableR-divisors in N1(Z/X). So KZ + ∆Z ∈ Mov(Z/X). Note that KZ +∆Z ∼R,g F and F is g-exceptional. By Lemma 2.4.4, −F is effective.

Anyway, there is i0 such that −Fi0 is effective. We put (Y,∆Y ) = (Zi0 ,∆Zi0

). Then this is a desired model. When(X,∆) is dlt, we can make a(E,X,∆) > −1 for every g-exceptionaldivisor by the definition of dlt pairs. In this case, f : Y → X isautomatically small by the above construction.

4.4. BCHM AND SOME RELATED RESULTS 119

Remark 4.4.22. It is conjectured that every minimal model pro-gram terminates. We can easily see that the minimal model programin the proof of Theorem 4.4.21 always terminates when (X,∆) is logcanonical. Note that Fi0 = 0 since Fi is always effective for every i.Therefore, KY + ∆Y = f ∗(KX + ∆) holds and is obviously f -nef when(X,∆) is log canonical.

4.4.23 (Infinitely many marked minimal models). The following ex-ample is due to Gongyo (see [G1]). For related examples, see Example4.5.12 and Example 4.5.9 below.

Example 4.4.24 (Infinitely many marked minimal models). Thereexists a three-dimensional projective plt pair (X,∆) with the followingproperties:

(i) KX + ∆ is nef and big, and(ii) there are infinitely many (KX + ∆)-flops.

Here we construct an example explicitly. We take a K3 surface S whichcontains infinitely many (−2)-curves. We take a projectively normalembedding S ⊂ PN . Let Z ⊂ PN+1 be a cone over S ⊂ PN and letϕ : X → Z be the blow-up at the vertex P of the cone Z. Thenthe projection Z 99K S from the vertex P induces a natural P1-bundlestructure p : X → S. Let E be the ϕ-exceptional divisor on X. ThenE is a section of p. In particular, E ' S. Note that

KX + E = ϕ∗KZ .

We take a sufficiently ample smooth Cartier divisor H on Z which doesnot pass through P . We further assume that KZ +H is ample. We put∆ = E + ϕ∗H and consider the pair (X,∆). By construction, (X,∆)is a plt threefold such that X is smooth and that KX + ∆ is big andsemi-ample. Since p : X → S is a P1-bundle and E is a section of p,we have

N1(X) = N1(E)⊕ R[l]

where l ' P1 is a fiber of p. Therefore, it is easy to see that

NE(E) ⊂ NE(X) ∩ (ϕ∗H = 0).

Claim. Let C be a (−2)-curve on E. Then R≥0[C] is an extremalray of NE(X) such that C · (KX + ∆) = 0.

Proof of Claim. Since C2 = −2 < 0, R≥0[C] is an extremal rayof NE(E). Let L be a supporting Cartier divisor of the extremal rayR≥0[C] ⊂ NE(E), that is,

NE(E) ∩ (L = 0) = R≥0[C].

120 4. MINIMAL MODEL PROGRAM

We can see that L is a Cartier divisor on S since S ' E. Then

(ϕ∗H + p∗L = 0) ∩NE(X) = R≥0[C].

Note that(KX + ∆) · C = KE · C = 0.

Thus R≥0[C] is an extremal ray of NE(X) with the desired intersectionnumber.

We put D = p∗(p(C)). Note that (X,∆ + δD) is plt for a smallpositive rational number δ. Then R≥0[C] is a (KX + ∆ + δD)-negativeextremal ray. Therefore, we obtain a (KX + ∆ + δD)-flip

(X,∆ + δD) //_______

&&LLLLLLLLLLL(X+,∆+ + δD+)

wwoooooooooooo

W

which is a (KX + ∆)-flop associated to the extremal ray R≥0[C]. Sincethere are infinitely many (−2)-curves on S, we obtain infinitely many(KX + ∆)-flops.

4.5. Fundamental theorems for normal pairs

In this section, we explain the main result of [F28] and some relatedresults and examples.

First, let us introduce the notion of normal pairs.

Definition 4.5.1 (Normal pairs). Let X be a normal algebraicvariety and let ∆ be an effective R-divisor on X such that KX + ∆ isR-Cartier. We call the pair (X,∆) a normal pair.

Next, we recall the main result of [F28], which covers the mainresult of [Am1] (see [Am1, Theorem 2]).

Theorem 4.5.2. Let X be a normal variety and let ∆ be an effectiveR-divisor such that KX + ∆ is R-Cartier, and let π : X → S be aprojective morphism onto a variety S. Then we have

NE(X/S) = NE(X/S)KX+∆≥0 +NE(X/S)Nlc(X,∆) +∑

Rj

with the following properties.

(1) Nlc(X,∆) is the non-lc locus of (X,∆) and

NE(X/S)Nlc(X,∆) = Im(NE(Nlc(X,∆)/S)→ NE(X/S)).

(2) Rj is a (KX+∆)-negative extremal ray of NE(X/S) such thatRj ∩NE(X/S)Nlc(X,∆) = 0 for every j.

4.5. FUNDAMENTAL THEOREMS FOR NORMAL PAIRS 121

(3) Let A be a π-ample R-divisor on X. Then there are onlyfinitely many Rj’s included in (KX + ∆ + A)<0. In partic-ular, the Rj’s are discrete in the half-space (KX + ∆)<0.

(4) Let F be a face of NE(X/S) such that

F ∩ (NE(X/S)KX+∆≥0 +NE(X/S)Nlc(X,∆)) = 0.

Then there exists a contraction morphism ϕF : X → Y overS.(i) Let C be an integral curve on X such that π(C) is a point.

Then ϕF (C) is a point if and only if [C] ∈ F .(ii) OY ' (ϕF )∗OX .(iii) Let L be a line bundle on X such that L ·C = 0 for every

curve C with [C] ∈ F . Then there is a line bundle LY onY such that L ' ϕ∗FLY .

(5) Every (KX + ∆)-negative extremal ray R with

R ∩NE(X/S)Nlc(X,∆) = 0

is spanned by a rational curve C with 0 < −(KX + ∆) · C ≤2 dimX.

From now on, we further assume that (X,∆) is log canonical, thatis, Nlc(X,∆) = ∅. Then we have the following properties.

(6) Let H be an effective R-Cartier R-divisor on X such that KX+∆ +H is π-nef and (X,∆ +H) is log canonical. Then, eitherKX +∆ is also π-nef or there is a (KX +∆)-negative extremalray R such that (KX + ∆ + λH) ·R = 0 where

λ := inft ≥ 0 |KX + ∆ + tH is π-nef .

Of course, KX + ∆ + λH is π-nef.

In [Am1], Ambro proved the properties (1), (2), (3), and (4) inTheorem 4.5.2 by using the theory of quasi-log schemes. More precisely,they are the main results of [Am1]. In [F28], the author obtainedTheorem 4.5.2 without using the theory of quasi-log schemes. Ourapproach in [F28] is much simpler than Ambro’s in [Am1]. For (5),see Theorem 4.6.7. For (6), see Theorem 4.7.3.

Let us include the following easy corollaries for the reader’s conve-nience.

Corollary 4.5.3 (cf. [KoMo, Corollary 3.17]). Let (X,∆) be alog canonical pair and let π : X → S be a projective morphism. LetR be a (KX + ∆)-negative extremal ray of NE(X/S) with contraction

122 4. MINIMAL MODEL PROGRAM

morphism ϕR : X → Y . Let C be a curve on X which generates R.Then we have an exact sequence

0 // Pic(Y )L7→ϕ∗

RL // Pic(X)M 7→(M ·C)

// Z.

In particular, we have ρ(Y/S) = ρ(X/S)− 1.

Proof. Let L be a line bundle on Y . Then (ϕR)∗(ϕ∗RL) = L.

Therefore, L 7→ ϕ∗RL is an injection. Note that M is a line bundle onX with (M · C) = 0 if and only if M = ϕ∗RL for some L by Theorem4.5.2 (4).

Corollary 4.5.4 (cf. [KoMo, Corollary 3.18]). Let (X,∆) be alog canonical pair and let π : X → S be a projective morphism. LetR be a (KX + ∆)-negative extremal ray of NE(X/S) with contractionmorphism ϕR : X → Y . Assume that X is Q-factorial and that ϕR iseither a divisorial or a Fano contraction. Then Y is also Q-factorial.

Proof. First, we assume that ϕR is divisorial. Let E be the ex-ceptional divisor on X. Then it is easy to see that (E · R) < 0 andthat E is irreducible. Let D be a Weil divisor on Y . Then there is arational number s such that

((ϕR)−1∗ D + sE ·R) = 0.

We take a positive integer m such that m((ϕR)−1∗ D + sE) is a Cartier

divisor on X. Then, by Theorem 4.5.2 (4), it is the pull-back of aCartier divisor DY on Y . Thus, mD ∼ DY . This implies that D isQ-Cartier.

Next, we assume that ϕR is a Fano contraction. Let D be a Weildivisor on Y . Let Y 0 be the smooth locus of Y . Let DX be the closureof (ϕR|ϕ−1

R (Y 0))∗(D|Y 0). Then DX is disjoint from the general fiber of

ϕR. Thus (DX ·R) = 0. We take a positive integer m such that mDX

is a Cartier divisor on X. Thus, by Theorem 4.5.2 (4), mDX ∼ ϕ∗RDY

for some Cartier divisor DY on Y . Thus, mD ∼ DY . This implies thatD is Q-Cartier.

Let us include the basepoint-free theorem for normal pairs in [F28]without proof for the reader’s convenience. Note that Theorem 4.5.5is a special case of Theorem 6.5.1 below.

Theorem 4.5.5 (see [F28, Theorem 13.1]). Let (X,∆) be a normalpair and let π : X → S be a projective morphism onto a variety S, andlet L be a π-nef Cartier divisor on X. Assume that

(i) aL− (KX + ∆) is π-ample for some real number a > 0, and(ii) ONlc(X,∆)(mL) is π|Nlc(X,∆)-generated for every m 0.

4.5. FUNDAMENTAL THEOREMS FOR NORMAL PAIRS 123

Then OX(mL) is π-generated for every m 0.

As an easy consequence of Theorem 4.5.5, we have:

Corollary 4.5.6. Let (X,∆) be a projective normal pair and letL be a nef Cartier divisor on X such that aL− (KX +∆) is nef and bigfor some real number a > 0. Assume that ONklt(X,∆)(mL) is generatedby global sections for every m 0. Then OX(mL) is generated byglobal sections for every m 0.

Proof. By Kodaira’s lemma (see Lemma 2.1.18), we can write

aL− (KX + ∆) ∼R A+ E

where A is an ample Q-divisor on X and E is an effective R-CartierR-divisor on X. Let ε be a small positive number. Then

Nklt(X,∆) = Nklt(X,∆ + εE)

scheme theoretically and aL−(KX+∆+εE) is ample. By replacing ∆with ∆+εE, we may assume that aL−(KX+∆) is ample. Since there isa natural surjective morphism ONklt(X,∆) → ONqlc(X,∆) , ONqlc(X,∆)(mL)is generated by global sections for everym 0. Therefore, by Theorem4.5.5, OX(mL) is generated by global sections for every m 0.

Note that it is well known that Corollary 4.5.6 can be proved by theusual X-method (see Section 4.2) with the aid of the Nadel vanishingtheorem (see Theorem 3.4.2).

4.5.7 (Examples of the Kleiman–Mori cone). From now on, we dis-cuss various examples of the Kleiman–Mori cone. The following exam-ple is well known (see, for example, [KMM, Example 4-2-4]).

Example 4.5.8. We take two smooth elliptic curves E1 and E2 onP2 such that P1 − P2 is not of finite order on the abelian group E1,where P1 and P2 are two of the nine intersection points of E1 and E2.Let f1 and f2 be the defining equations of E1 and E2 respectively. Therational map which maps x ∈ P2 \ (E1 ∩ E2) to (f1(x) : f2(x)) ∈ P1

becomes a morphism from S which is obtained by taking blow-ups of P2

at the nine intersection points of E1 and E2. Then it is easy to see thatthe inverse images of P1 and P2 on S are sections of π : S → P1. Bythe choice of P1 and P2, there are infinitely many sections of π, whichare (−1)-curves. Therefore, NE(S) has infinitely many KS-negativeextremal rays.

Example 4.5.9, which is essentially the same as [G2, Example 5.6],is an answer to [KMM, Problem 4-2-5]. Although the construction isessentially the same as that of Example 4.4.24, we explain the detailsof the construction for the reader’s convenience.

124 4. MINIMAL MODEL PROGRAM

Example 4.5.9 (Infinitely many flipping contractions). There ex-ists a three-dimensional projective plt pair (X,∆) with the followingproperties:

(i) KX + ∆ is big, and(ii) there are infinitely many (KX + ∆)-negative extremal rays.

Here we construct an example explicitly. Let S be a rational ellipticsurface with infinitely many (−1)-curves constructed in Example 4.5.8.We take a projectively normal embedding S ⊂ PN . Let Z ⊂ PN+1 bea cone over S ⊂ PN and let ϕ : X → Z be the blow-up at the vertex Pof the cone Z. Then the projection Z 99K S from the vertex P inducesa natural P1-bundle structure p : X → S. Let E be the ϕ-exceptionaldivisor on X. Then E is a section of p. In particular, E ' S. Wetake a sufficiently ample smooth divisor H on Z which does not passthrough P . We put ∆ = E + ϕ∗H and consider the pair (X,∆). Bythe construction, (X,∆) is a plt threefold such that X is smooth andthat KX + ∆ is big. Since p : X → S is a P1-bundle and E is a sectionof p, we have

N1(X) = N1(E)⊕ R[l]

where l ' P1 is a fiber of p. Therefore, it is easy to see that

NE(E) ⊂ NE(X) ∩ (ϕ∗H = 0).

Claim. Let C be a (−1)-curve on E. Then R≥0[C] is a (KX +∆)-negative extremal ray of NE(X).

Proof of Claim. Note that R≥0[C] is a KE-negative extremalray of NE(E). Let L be a supporting Cartier divisor of R≥0[C] ⊂NE(E). We can see that L is a Cartier divisor on S since S ' E.Then

(ϕ∗H + p∗L = 0) ∩NE(X) = R≥0[C].

Note that(KX + ∆) · C = KE · C = −1.

Thus R≥0[C] is a (KX + ∆)-negative extremal ray of NE(X). Therefore, there are infinitely many (KX + ∆)-negative extremal

rays of NE(X). Note that every extremal ray corresponds to a flippingcontraction with respect to KX + ∆.

Remark 4.5.10. Let (X,∆) be a projective klt pair. Assume thatKX + ∆ is big. Then there are only finitely many (KX + ∆)-negativeextremal rays. We can check this well-known result as follows. ByKodaira’s lemma (see Lemma 2.1.18), we can write

KX + ∆ ∼R A+ E

4.5. FUNDAMENTAL THEOREMS FOR NORMAL PAIRS 125

where A is an ample Q-divisor on X and E is an effective R-Cartier R-divisor on X. Let ε be a small rational number such that (X,∆ + εE)is klt. In this case, (KX + ∆)-negative extremal ray is nothing but(KX + ∆ + εE + εA)-negative extremal ray. By Theorem 4.5.2 (3),there are only finitely many (KX + ∆)-negative extremal rays.

Lemma 4.5.11. Let (X,∆) be a Q-factorial projective log canonicalpair such that KX + ∆ ∼R D ≥ 0. Then there are only finitely many(KX + ∆)-negative extremal rays inducing divisorial contractions. Inparticular, if X is a smooth projective threefold with κ(X,KX) ≥ 0,then there are only finitely many KX-negative extremal rays.

Proof. Let R be a (KX + ∆)-negative extremal ray such that theassociated contraction ϕR : X → Y is divisorial. Then the exceptionallocus of ϕR is a prime divisor E onX which is an irreducible componentof SuppD. Therefore, there are only finitely many (KX + ∆)-negativedivisorial contractions. When X is a smooth projective threefold withκ(X,KX) ≥ 0, the contraction morphism ϕR : X → Y is divisorialfor every KX-negative extremal ray R by [Mo2] (see Theorem 1.1.4).Therefore, there are only finitely many KX-negative extremal rays fora smooth projective threefold X with κ(X,KX) ≥ 0.

Yoshinori Gongyo and Yoshinori Namikawa informed the authorof the following example. It is well known as Schoen’s Calabi–Yauthreefold and is an answer to [KMM, Problem 4-2-5].

Example 4.5.12. Let π1 : S1 → P1 and π2 : S2 → P1 be ratio-nal elliptic surfaces with infinitely many (−1)-curves constructed inExample 4.5.8. We put X = S1 ×P1 S2.

S1 ×P1 S2

p1

zzuuuuuuuuuup2

$$IIIIIIIIII

S1

π1$$JJJJJJJJJJJ S2

π2zzttttttttttt

P1

We assume that π−11 (p) or π−1

2 (p) is smooth for every point p ∈ P1.Then it is easy to see that X is a smooth projective threefold withKX ∼ 0 by using the canonical bundle formula for rational ellipticsurfaces (see [Scho] and [BHPV, Chapter V. (12.3) Corollary]). Wecan directly check H1(X,OX) = H2(X,OX) = 0. Therefore, X is aCalabi–Yau threefold. Let l be a (−1)-curve on S1 and let mλλ∈Λ be

126 4. MINIMAL MODEL PROGRAM

the set of all (−1)-curves on S2. Then Cλ = l ×P1 mλ is a (−1,−1)-curve, that is, a rational curve whose normal bundle is isomorphicto OP1(−1) ⊕ OP1(−1), on X for every λ ∈ Λ. We take a semi-ampleCartier divisorH on S1 which is a supporting Cartier divisor of R≥0[l] ⊂NE(S1). Let Hλ be a semi-ample Cartier divisor on S2 which is asupporting Cartier divisor of R≥0[mλ] ⊂ NE(S2) for every λ ∈ Λ.Then p∗1H + p∗2Hλ induces a contraction morphism ϕλ : X → Wλ suchthat Exc(ϕλ) = Cλ for every λ ∈ Λ. Therefore, R≥0[Cλ] is an extremalray of NE(X). We put D = l ×P1 S2. Then it is easy to see that(KX + εD) · Cλ = −ε for every λ ∈ Λ. Therefore, (X, εD) is a kltthreefold which has infinitely many (KX + εD)-negative extremal raysfor 0 < ε 1. Note that we have the following flopping diagram

X

ϕλ @@@

@@@@

@φλ //________ X+

λ

ϕ+λ

Wλ

where X+λ is a smooth projective threefold with KX+

λ∼ 0. Although

we have infinitely many flops φλ : X 99K X+λ , Namikawa (see [Nam])

proved that there are only finitely many X+λ up to isomorphisms. For

the details, see [Nam].

4.6. Lengths of extremal rays

In this section, which is essentially a reproduction of [F28, Section18], we discuss estimates of lengths of extremal rays. It is indispensablefor the log minimal model program with scaling (see, for example,[BCHM]) and the geography of log models (see, for example, [Sh3]and [ShCh]). The results in this section were obtained in [Ko4], [Ko5],and [Ka4], [Sh3], [Sh5], and [Bir2] with some extra assumptions.

Let us recall the following easy lemma.

Lemma 4.6.1 (cf. [Sh5, Lemma 1]). Let (X,∆) be a log canonicalpair, where ∆ is an R-divisor. Then there are positive real numbersri, effective Q-divisors ∆i for 1 ≤ i ≤ l, and a positive integer m suchthat

∑li=1 ri = 1,

KX + ∆ =l∑

i=1

ri(KX + ∆i),

(X,∆i) is log canonical for every i, and m(KX + ∆i) is Cartier forevery i.

4.6. LENGTHS OF EXTREMAL RAYS 127

Proof. Let∑

kDk be the irreducible decomposition of Supp ∆.We consider the finite dimensional real vector space V =

⊕k

RDk. We

put

Q = D ∈ V | KX +D is R-Cartier .

Then, it is easy to see that Q is an affine subspace of V defined overQ. We put

L = D ∈ Q | KX +D is log canonical .

Thus, by the definition of log canonicity, it is also easy to check that Lis a closed convex rational polytope in V . We note that L is compact inthe classical topology of V . By assumption, ∆ ∈ L. Therefore, we canfind the desired Q-divisors ∆i ∈ L and positive real numbers ri.

The next result is essentially due to [Ka4] and [Sh5, Proposition1]. We will prove a more general result in Theorem 4.6.7 whose proofdepends on Theorem 4.6.2.

Theorem 4.6.2. Let X be a normal variety such that (X,∆) is logcanonical and let π : X → S be a projective morphism onto a varietyS. Let R be a (KX + ∆)-negative extremal ray. Then we can find a(possibly singular) rational curve C on X such that [C] ∈ R and

0 < −(KX + ∆) · C ≤ 2 dimX.

Proof. By shrinking S, we may assume that S is quasi–projective.By replacing π : X → S with the extremal contraction ϕR : X → Yover S (see Theorem 4.5.2 (4)), we may assume that the relative Picardnumber ρ(X/S) = 1. In particular, −(KX + ∆) is π-ample. Let

KX + ∆ =l∑

i=1

ri(KX + ∆i)

be as in Lemma 4.6.1. We assume that −(KX + ∆1) is π-ample and−(KX + ∆i) = −si(KX + ∆1) in N1(X/S) with si ≤ 1 for every i ≥ 2.Thus, it is sufficient to find a rational curve C such that π(C) is a pointand that

−(KX + ∆1) · C ≤ 2 dimX.

So, we may assume that KX + ∆ is Q-Cartier and log canonical. Bytaking a dlt blow-up (see Theorem 4.4.21), there is a birational mor-phism f : (Y,∆Y ) → (X,∆) such that KY + ∆Y = f∗(KX + ∆), Y isQ-factorial, and (Y,∆Y ) is dlt. By [Ka4, Theorem 1] and [Ma, Theo-rem 10-2-1] (see also [Deb, Section 7.11]), we can find a rational curve

128 4. MINIMAL MODEL PROGRAM

C ′ on Y such that

−(KY + ∆Y ) · C ′ ≤ 2 dimY = 2 dimX

and that C ′ spans a (KY + ∆Y )-negative extremal ray. By the pro-jection formula, the f -image of C ′ is a desired rational curve. So, wefinish the proof.

Remark 4.6.3. It is conjectured that the estimate ≤ 2 dimX inTheorem 4.6.2 should be replaced by ≤ dimX+1. When X is smoothprojective, it is true by Mori’s famous result (see [Mo2], Theorem 1.1.1,and [KoMo, Theorem 1.13]). When X is a toric variety, it is also trueby [F4] and [F10].

Remark 4.6.4. In the proof of Theorem 4.6.2, we need Kawamata’sestimate on the length of an extremal rational curve (see, for example,[Ka4, Theorem 1], [Ma, Theorem 10-2-1], and [Deb, Section 7.11]). Itdepends on Mori’s bend and break technique to create rational curves.So, we need the mod p reduction technique there.

Remark 4.6.5. Let (X,D) be a log canonical pair such that D isan R-divisor. Let φ : X → Y be a projective morphism and let Hbe a Cartier divisor on X. Assume that H − (KX + D) is f -ample.By the Kawamata–Viehweg type vanishing theorem for log canonicalpairs (see Theorem 5.6.4), Rqφ∗OX(H) = 0 for every q > 0 if X andY are algebraic varieties. If this vanishing theorem holds for analyticspaces X and Y , then Kawamata’s original argument in [Ka4] worksdirectly for log canonical pairs. In that case, we do not need dlt blow-ups (see Theorem 4.4.21), which follows from [BCHM], in the proofof Theorem 4.6.2.

We consider the proof of [Ma, Theorem 10-2-1] when (X,D) isQ-factorial dlt. We need R1φ∗OX(H) = 0 after shrinking X and Yanalytically. In our situation, (X,D − εbDc) is klt for 0 < ε 1.Therefore, H − (KX +D− εbDc) is φ-ample and (X,D− εbDc) is kltfor 0 < ε 1. Thus, we can apply the analytic version of the relativeKawamata–Viehweg vanishing theorem (see, for example, [F31]). So,we do not need the analytic version of the Kawamata–Viehweg typevanishing theorem for log canonical pairs.

Remark 4.6.6. We give a remark on [BCHM]. We use the samenotation as in [BCHM, 3.8]. In the proof of [BCHM, Corollary 3.8.2],we may assume that KX + ∆ is klt by [BCHM, Lemma 3.7.4]. Byperturbing the coefficients of B slightly, we can further assume thatB is a Q-divisor. By applying the usual cone theorem to the klt pair(X,B), we obtain that there are only finitely many (KX +∆)-negative

4.6. LENGTHS OF EXTREMAL RAYS 129

extremal rays of NE(X/U). We note that [BCHM, Theorem 3.8.1]is only used in the proof of [BCHM, Corollary 3.8.2]. Therefore, wedo not need the estimate of lengths of extremal rays in [BCHM]. Inparticular, we do not need mod p reduction arguments for the proof ofthe main results in [BCHM].

The final result in this section is an estimate of lengths of ex-tremal rays which are relatively ample at non-lc loci (see also [Ko4]and [Ko5]).

Theorem 4.6.7 (Theorem 4.5.2 (5)). Let X be a normal variety,let ∆ be an effective R-divisor on X such that KX + ∆ is R-Cartier,and let π : X → S be a projective morphism onto a variety S. Let Rbe a (KX + ∆)-negative extremal ray of NE(X/S) which is relativelyample at Nlc(X,∆), that is, R∩NE(X/S)Nlc(X,∆) = 0. Then we canfind a (possibly singular) rational curve C on X such that [C] ∈ R and

0 < −(KX + ∆) · C ≤ 2 dimX.

Proof. By shrinking S, we may assume that S is quasi-projective.By replacing π : X → S with the extremal contraction ϕR : X → Yover S (see Theorem 4.5.2 (4)), we may assume that the relative Picardnumber ρ(X/S) = 1 and that π is an isomorphism in a neighborhoodof Nlc(X,∆). In particular, −(KX + ∆) is π-ample. By taking a dltblow-up (see Theorem 4.4.21), there is a projective birational morphismf : Y → X such that

(i) KY +∆Y = f ∗(KX +∆)+∑

a(E,X,∆)<−1

(a(E,X,∆)+1)E, where

∆Y = f−1∗ ∆ +

∑E:f -exceptional

E,

(ii) (Y,∆Y ) is a Q-factorial dlt pair, and(iii) KY +D = f ∗(KX + ∆) with D = ∆Y + F , where

F = −∑

a(E,X,∆)<−1

(a(E,X,∆) + 1)E ≥ 0.

Therefore, we have

f∗(NE(Y/S)KY +D≥0) ⊆ NE(X/S)KX+∆≥0 = 0.We also note that

f∗(NE(Y/S)Nlc(Y,D)) = 0.Thus, there is a (KY +D)-negative extremal ray R′ of NE(Y/S) whichis relatively ample at Nlc(Y,D). By Theorem 4.5.2, R′ is spanned bya curve C†. Since −(KY +D) ·C† > 0, we see that f(C†) is a curve. If

130 4. MINIMAL MODEL PROGRAM

C† ⊂ SuppF , then f(C†) ⊂ Nlc(X,∆). This is a contradiction becauseπ f(C†) is a point. Thus, C† 6⊂ SuppF . Since

−(KY + ∆Y ) = −(KY +D) + F,

we can see that R′ is a (KY +∆Y )-negative extremal ray of NE(Y/S).Therefore, we can find a rational curve C ′ on Y such that C ′ spans R′

and that0 < −(KY + ∆Y ) · C ′ ≤ 2 dimX

by Theorem 4.6.2. By the above argument, we can easily see thatC ′ 6⊂ SuppF . Therefore, we obtain

0 < −(KY +D) · C ′ = −(KY + ∆Y ) · C ′ − F · C ′

≤ −(KY + ∆Y ) · C ′ ≤ 2 dimX.

Since KY +D = f ∗(KX +∆), C = f(C ′) is a rational curve on X suchthat π(C) is a point and 0 < −(KX + ∆) · C ≤ 2 dimX.

Remark 4.6.8. In Theorem 4.6.7, we can prove 0 < −(KX + ∆) ·C ≤ dimX+1 when dimX ≤ 2. For the details, see [F29, Proposition3.7].

4.7. Shokurov polytope

In this section, we discuss a very important result obtained byShokurov (cf. [Sh3, 6.2. First Main Theorem]), which is an applica-tion of Theorem 4.6.2. We closely follow Birkar’s treatment in [Bir3,Section 3].

4.7.1. Let π : X → S be a projective morphism from a normalvariety X to a variety S. A curve Γ on X is called extremal over S ifthe following properties hold.

(1) Γ generates an extremal ray R of NE(X/S).(2) There is a π-ample Cartier divisor H on X such that

H · Γ = minH · C,where C ranges over curves generating R.

We note that every (KX + ∆)-negative extremal ray R of NE(X/S)is spanned by a curve if ∆ is an effective R-divisor on X such that(X,∆) is log canonical. It is a consequence of the cone and contractiontheorem (see Theorem 4.5.2).

Let ∆ be an effective R-divisor on X such that (X,∆) is log canoni-cal and let R be a (KX +∆)-negative extremal ray of NE(X/S). Thenwe can take a rational curve C such that C spans R and that

0 < −(KX + ∆) · C ≤ 2 dimX

4.7. SHOKUROV POLYTOPE 131

by Theorem 4.6.2. Let Γ be an extremal curve generating R. Then wehave

−(KX + ∆) · ΓH · Γ

=−(KX + ∆) · C

H · C.

Therefore,

−(KX + ∆) · Γ = (−(KX + ∆) · C) · H · ΓH · C

≤ 2 dimX.

Let F be a reduced divisor onX. We consider the finite dimensionalreal vector space V =

⊕k RFk where F =

∑k Fk is the irreducible

decomposition. We have already seen that

L = D ∈ V | (X,D) is log canonical

is a rational polytope in V , that is, it is the convex hull of finitely manyrational points in V (see Lemma 4.6.1).

Let D1, · · · , Dr be the vertices of L and let m be a positive integersuch that m(KX + Dj) is Cartier for every j. We take an R-divisor∆ ∈ L. Then we can find non-negative real numbers a1, · · · , ar suchthat ∆ =

∑j ajDj,

∑j aj = 1, and (X,Dj) is log canonical for every j

(see Lemma 4.6.1). For every curve C on X, the intersection number−(KX + ∆) · C can be written as∑

j

ajnjm

such that nj ∈ Z for every j. If C is an extremal curve, then we cansee that nj ≤ 2m dimX for every j by the above arguments.

On the real vector space V , we consider the following norm

||∆|| = maxj|bj|,

where ∆ =∑

j bjFj.

We explain Shokurov’s important results (cf. [Sh3]) following [Bir3,Proposition 3.2].

Theorem 4.7.2. We use the same notation as in 4.7.1. We fix anR-divisor ∆ ∈ L. Then we can find positive real numbers α and δ,which depend on (X,∆) and F , with the following properties.

(1) If Γ is any extremal curve over S and (KX + ∆) · Γ > 0, then(KX + ∆) · Γ > α.

(2) If D ∈ L, ||D−∆|| < δ, and (KX +D) ·R ≤ 0 for an extremalcurve Γ, then (KX + ∆) · Γ ≤ 0.

132 4. MINIMAL MODEL PROGRAM

(3) Let Rtt∈T be any set of extremal rays of NE(X/S). Then

NT = D ∈ L | (KX +D) ·Rt ≥ 0 for every t ∈ T

is a rational polytope in V .

Proof. (1) If ∆ is a Q-divisor, then the claim is obvious even if Γis not extremal. We assume that ∆ is not a Q-divisor. Then we canwrite KX + ∆ =

∑j aj(KX + Dj) as in 4.7.1. Then (KX + ∆) · Γ =∑

j aj(KX +Dj) · Γ. If (KX + ∆) · Γ < 1, then

−2 dimX ≤ (KX +Dj0) · Γ <1

aj0−

∑j 6=j0

aj(KX +Dj) · Γ + 1

≤ 2 dimX + 1

aj0

for aj0 6= 0. This is because (KX + Dj) · Γ ≥ −2 dimX for everyj. Thus there are only finitely many possibilities of the intersectionnumbers (KX +Dj) · Γ for aj 6= 0 when (KX + ∆) · Γ < 1. Therefore,the existence of α is obvious.

(2) If we take δ sufficiently small, then, for every D ∈ L with||D −∆|| < δ, we can always find D′ ∈ L such that

KX +D = (1− s)(KX + ∆) + s(KX +D′)

with

0 ≤ s ≤ α

α+ 2 dimX.

Since Γ is extremal, we have (KX+D′)·Γ ≥ −2 dimX for everyD′ ∈ L.We assume that (KX + ∆) · Γ > 0. Then (KX + ∆) · Γ > α by (1).Therefore,

(KX +D) · Γ = (1− s)(KX + ∆) · Γ + s(KX +D′) · Γ> (1− s)α+ s(−2 dimX) ≥ 0.

This is a contradiction. Therefore, we obtain (KX + ∆) · Γ ≤ 0. Wecomplete the proof of (2).

(3) For every t ∈ T , we may assume that there is some Dt ∈ Lsuch that (KX + Dt) · Rt < 0. We note that (KX + D) · Rt < 0 forsome D ∈ L implies (KX + Dj) · Rt < 0 for some j. Therefore, wemay assume that T is contained in N. This is because there are onlycountably many (KX +Dj)-negative extremal rays for every j by thecone theorem (see Theorem 4.5.2). We note that NT is a closed convexsubset of L by definition. If T is a finite set, then the claim is obvious.Thus, we may assume that T = N. By (2) and by the compactness of

4.7. SHOKUROV POLYTOPE 133

NT , we can take ∆1, · · · ,∆n ∈ NT and δ1, · · · , δn > 0 such that NT iscovered by

Bi = D ∈ L | ||D −∆i|| < δiand that if D ∈ Bi with (KX +D) ·Rt < 0 for some t, then (KX +∆i) ·Rt = 0. If we put

Ti = t ∈ T | (KX +D) ·Rt < 0 for some D ∈ Bi,then (KX + ∆i) · Rt = 0 for every t ∈ Ti by the above construction.Since Bini=1 gives an open covering of NT , we have NT =

∩1≤i≤nNTi

by the following claim.

Claim. NT =∩

1≤i≤nNTi.

Proof of Claim. We note that NT ⊂∩

1≤i≤nNTiis obvious. We

assume that NT (∩

1≤i≤nNTi. We take D ∈

∩1≤i≤nNTi

\ NT whichis very close to NT . Since NT is covered by Bini=1, there is some i0such that D ∈ Bi0 . Since D 6∈ NT , there is some t0 ∈ T such that(KX + D) · Rt0 < 0. Thus, t0 ∈ Ti0 . This is a contradiction becauseD ∈ NTi0

. Therefore, NT =∩

1≤i≤nNTi.

So, it is sufficient to see that each NTiis a rational polytope in V .

By replacing T with Ti, we may assume that there is some D ∈ NTsuch that (KX +D) ·Rt = 0 for every t ∈ T .

If dimR L = 1, then this already implies the claim. We assumedimR L > 1. Let L1, · · · ,Lp be the proper faces of L. Then N i

T =NT ∩ Li is a rational polytope by induction on dimension. Moreover,for each D′′ ∈ NT which is not D, there is D′ on some proper face ofL such that D′′ is on the line segment determined by D and D′. Notethat (KX + D) · Rt = 0 for every t ∈ T . Therefore, if D′ ∈ Li, thenD′ ∈ N i

T . Thus, NT is the convex hull of D and all the N iT . So there

is a finite subset T ′ ⊂ T such that∪i

N iT = NT ′ ∩ (

∪i

Li).

Therefore, the convex hull of D and∪iN i

T is just NT ′ . We completethe proof of (3).

By Theorem 4.7.2 (3), Lemma 2.6 in [Bir2] holds for log canonicalpairs. It may be useful for the minimal model program with scaling.

Theorem 4.7.3 (cf. [Bir2, Lemma 2.6]). Let (X,∆) be a log canon-ical pair, let ∆ be an R-divisor, and let π : X → S be a projectivemorphism between algebraic varieties. Let H be an effective R-CartierR-divisor on X such that KX + ∆ +H is π-nef and (X,∆ +H) is log

134 4. MINIMAL MODEL PROGRAM

canonical. Then, either KX + ∆ is also π-nef or there is a (KX + ∆)-negative extremal ray R such that (KX + ∆ + λH) ·R = 0, where

λ := inft ≥ 0 |KX + ∆ + tH is π-nef .Of course, KX + ∆ + λH is π-nef.

Note that Theorem 4.7.3 is nothing but Theorem 4.5.2 (6).

Proof. Assume that KX + ∆ is not π-nef. Let Rj be the set of(KX + ∆)-negative extremal rays over S. Let Cj be an extremal curvespanning Rj for every j. We put µ = sup

jµj, where

µj =−(KX + ∆) · Cj

H · Cj.

Obviously, λ = µ and 0 < µ ≤ 1. So, it is sufficient to prove thatµ = µj0 for some j0. There are positive real numbers r1, · · · , rl suchthat

∑i ri = 1 and a positive integer m, which are independent of j,

such that

−(KX + ∆) · Cj =l∑

i=1

rinijm

> 0

(see Lemma 4.6.1, Theorem 4.6.2, and 4.7.1). Since Cj is extremal, nijis an integer with nij ≤ 2m dimX for every i and j. If (KX + ∆ +H) · Rj0 = 0 for some j0, then there are nothing to prove since λ = 1and (KX + ∆ + H) · R = 0 with R = Rj0 . Thus, we assume that(KX + ∆ + H) · Rj > 0 for every j. We put F = Supp(∆ + H). LetF =

∑k Fk be the irreducible decomposition. We put V =

⊕k RFk,

L = D ∈ V | (X,D) is log canonical,and

N = D ∈ L | (KX +D) ·Rj ≥ 0 for every j.Then N is a rational polytope in V by Theorem 4.7.2 (3) and ∆ +His in the relative interior of N by the above assumption. Therefore, wecan write

KX + ∆ +H =

q∑p=1

r′p(KX +Dp),

where r′1, · · · , r′q are positive real numbers such that∑

p r′p = 1, (X,Dp)

is log canonical for every p, m′(KX +Dp) is Cartier for some positiveinteger m′ and every p, and (KX +Dp) · Cj > 0 for every p and j. So,we obtain

(KX + ∆ +H) · Cj =

q∑p=1

r′pn′pj

m′

4.7. SHOKUROV POLYTOPE 135

with 0 < n′pj = m′(KX + Dp) · Cj ∈ Z. Note that m′ and r′p areindependent of j for every p. We also note that

1

µj=

H · Cj−(KX + ∆) · Cj

=(KX + ∆ +H) · Cj−(KX + ∆) · Cj

+ 1

=m

∑qp=1 r

′pn′pj

m′∑l

i=1 rjnij+ 1.

Sincel∑

i=1

rinijm

> 0

for every j and nij ≤ 2m dimX with nij ∈ Z for every i and j, thenumber of the set niji,j is finite. Thus,

infj

1

µj

=

1

µj0for some j0. Therefore, we obtain µ = µj0 . We finish the proof.

Let us recall the abundance conjecture, which is one of the most im-portant conjectures in the minimal model theory for higher-dimensionalalgebraic varieties.

4.7.4 (Abundance conjecture). We treat some applications of The-orem 4.7.2 (3) to the abundance conjecture for R-divisors (see [Sh3,2.7. Theorem on log semi-ampleness for 3-folds]).

Conjecture 4.7.5 (Abundance conjecture). Let (X,∆) be a logcanonical pair and let f : X → Y be a projective morphism betweenvarieties. If KX + ∆ is f -nef, then KX + ∆ is f -semi-ample.

For the recent developments of the abundance conjecture, see, forexample, [FG1].

The following proposition is a useful application of Theorem 4.7.2(see [Sh3, 2.7]).

Proposition 4.7.6. Let f : X → Y be a projective morphismbetween algebraic varieties. Let ∆ be an effective R-divisor on X suchthat (X,∆) is log canonical and that KX +∆ is f -nef. Assume that theabundance conjecture holds for Q-divisors. More precisely, we assumethat KX +D is f -semi-ample if D ∈ L, D is a Q-divisor, and KX +Dis f -nef, where

L = D ∈ V | (X,D) is log canonical,V =

⊕k RFk, and

∑k Fk is the irreducible decomposition of Supp ∆.

Then KX + ∆ is f -semi-ample.

136 4. MINIMAL MODEL PROGRAM

Proof. Let Rtt∈T be the set of all extremal rays of NE(X/Y ).We consider NT as in Theorem 4.7.2 (3). Then NT is a rational poly-tope in L by Theorem 4.7.2 (3). We can easily see that

NT = D ∈ L |KX +D is f -nef.By assumption, ∆ ∈ NT . Let F be the minimal face of NT containing∆. Then we can find Q-divisors D1, · · · , Dl on X such that Di is inthe relative interior of F ,

KX + ∆ =∑i

di(KX +Di),

where di is a positive real number for every i and∑

i di = 1. Byassumption, KX +Di is f -semi-ample for every i. Therefore, KX + ∆is f -semi-ample.

Remark 4.7.7 (Stability of Iitaka fibrations). In the proof of Propo-sition 4.7.6, we note the following property. If C is a curve on Xsuch that f(C) is a point and (KX + Di0) · C = 0 for some i0, then(KX +Di) ·C = 0 for every i. This is because we can find ∆′ ∈ F suchthat (KX+∆′)·C < 0 if (KX+Di)·C > 0 for some i 6= i0. This is a con-tradiction. Therefore, there exists a contraction morphism g : X → Zover Y and h-ample Q-divisors A1, · · · , Al on Z, where h : Z → Y ,such that KX +Di ∼Q g

∗Ai for every i. In particular,

KX + ∆ ∼R g∗(

∑i

diAi).

Note that∑

i diAi is h-ample. Roughly speaking, the Iitaka fibrationof KX + ∆ is the same as that of KX +Di for every i.

Corollary 4.7.8. Let f : X → Y be a projective morphism be-tween algebraic varieties. Assume that (X,∆) is log canonical and thatKX + ∆ is f -nef. We further assume one of the following conditions.

(i) dimX ≤ 3.(ii) dimX = 4 and dimY ≥ 1.

Then KX + ∆ is f -semi-ample.

Proof. It is obvious by Proposition 4.7.6 and the log abundancetheorems for threefolds and fourfolds (see, for example, [KeMM, 1.1. The-orem] and [F22, Theorem 3.10]).

Corollary 4.7.9. Let f : X → Y be a projective morphism be-tween algebraic varieties. Assume that (X,∆) is klt and KX + ∆ isf -nef. We further assume that dimX − dimY ≤ 3. Then KX + ∆ isf -semi-ample.

4.8. MMP FOR LC PAIRS 137

Proof. If ∆ is a Q-divisor, then it is well known that KXη + ∆η

is semi-ample, where Xη is the generic fiber of f and ∆η = ∆|Xη (see,for example, [KeMM, 1.1. Theorem]). Therefore, KX + ∆ is f -semi-ample by [F24, Theorem 1.1]. When ∆ is an R-divisor, we can takeQ-divisors D1, · · · , Dl ∈ F as in the proof of Proposition 4.7.6 suchthat (X,Di) is klt for every i. Since KX +Di is f -semi-ample by theabove argument, we obtain that KX + ∆ is f -semi-ample.

4.8. MMP for lc pairs

In this section, we discuss the minimal model program for log canon-ical pairs and some related topics.

Let us start with the definition of log canonical models.

Definition 4.8.1 (Log canonical model). Let (X,∆) be a log canon-ical pair and let π : X → S be a proper morphism. A pair (X ′,∆′)sitting in a diagram

(X,∆)φ //_______

π""F

FFFF

FFFF

(X ′,∆′)

π′wwwwwwwww

S

is called a log canonical model of (X,∆) over S if

(i) π′ is proper,(ii) φ−1 has no exceptional divisors,(iii) ∆′ = φ∗∆,(iv) KX′ + ∆′ is π′-ample, and(v) a(E,X,∆) ≤ a(E,X ′,∆′) for every φ-exceptional divisor E ⊂

X.

Lemma 4.8.2. Let (X,∆) be a log canonical pair and let π : X → Sbe a proper morphism. Let (X ′,∆′) be a log canonical model of (X,∆)over S. Then a(E,X,∆) ≤ a(E,X ′,∆′) for every prime divisor Eover X. We assume that ∆ is a Q-divisor. Then

X ′ = ProjX⊕m≥0

π∗OX(bm(KX + ∆)c).

138 4. MINIMAL MODEL PROGRAM

Proof. By the same argument as Lemma 4.3.2, we have a(E,X,∆) ≤a(E,X ′,∆′) for every prime divisor E over X. We take a common res-olution

Wq

!!BBB

BBBB

Bp

~~

X

π A

AAAA

AAA

φ //_______ X ′

π′||

||||

||

S

of X and X ′. We put ∆W = p−1∗ ∆ + E, where E is the sum of all

p-exceptional divisors. Then we can write

KW + ∆W = p∗(KX + ∆) + F

and

KW + ∆W = q∗(KX′ + ∆′) +G

where F is effective and p-exceptional, andG is effective and q-exceptional.Therefore,

X ′ = ProjS⊕m≥0

π′∗OX′(bm(KX′ + ∆′)c)

= ProjS⊕m≥0

(π′ q)∗OW (bm(KW + ∆W )c)

= ProjS⊕m≥0

π∗OX(bm(KX + ∆)c).

This is the desired description of X ′.

Lemma 4.8.3. Let (X,∆) be a log canonical pair and let π : X → Sbe a proper morphism onto a variety. Let (Xm,∆m) be a minimal modelof (X,∆) over S and let (X lc,∆lc) be a log canonical model of (X,∆)over S. Then there is a natural morphism α : Xm → X lc such that

KXm + ∆m = α∗(KXlc + ∆lc).

In particular, KXm + ∆m is semi-ample over S, that is, (Xm,∆m) is agood minimal model of (X,∆) over S.

Proof. We take a common resolution

X Wpoo r //

q

X lc

Xm

4.8. MMP FOR LC PAIRS 139

of X, Xm, and X lc. Let E be the sum of all p-exceptional divisors. Weput ∆W = p−1

∗ ∆ + E. Then we have

KW + ∆W = q∗(KXm + ∆m) + F

andKW + ∆W = r∗(KXlc + ∆lc) +G

where F is effective and q-exceptional andG is effective and r-exceptionalby the negativity lemma (see Lemma 2.3.26). Therefore, we obtain

q∗(KXm + ∆m) + F = r∗(KXlc + ∆lc) +G.

Note that q∗(G − F ) is effective and −(G − F ) is q-nef. This impliesG − F ≥ 0 by the negativity lemma (see Lemma 2.3.26). Similarly,r∗(F −G) is effective and −(F −G) is r-nef. This implies F −G ≥ 0by the negativity lemma (see Lemma 2.3.26) again. Therefore, F = G.So, we have

q∗(KXm + ∆m) = r∗(KXlc + ∆lc).

We assume that r q−1 : Xm 99K X lc is not a morphism. Then wecan find a curve C on W such that q(C) is a point and that r(C) is acurve. In this case,

0 = C · q∗(KXm + ∆m) = C · r∗(KXlc + ∆lc) > 0.

This is a contradiction. Therefore, α : r q−1 : Xm 99K X lc is amorphism and KXm + ∆m = α∗(KXlc + ∆lc).

By the proof of Lemma 4.8.3, we have:

Corollary 4.8.4. Let (X,∆) be a log canonical pair and let π :X → S be a proper morphism onto a variety. Let (X1,∆1) and (X2,∆2)be log canonical models of (X,∆) over S. Then (X1,∆1) is isomorphicto (X2,∆2) over S. Therefore, the log canonical model of (X,∆) overS is unique.

In order to discuss the minimal model program for log canonicalpairs, it is convenient to use the following definitions of minimal modelsand Mori fiber spaces due to Birkar–Shokurov.

Definition 4.8.5 (Minimal models, see [Bir4, Definition 2.1]).Let (X,∆) be a log canonical pair and let π : X → S be a projectivemorphism onto a variety S. let φ : X 99K Y be a birational map over

S. We put ∆Y = ∆ + E where ∆ is the birational transform of ∆ onY and E is the reduced exceptional divisor of φ−1, that is, E =

∑j Ej

where Ej is a prime divisor on Y which is exceptional over X for everyj. We assume that

(i) (Y,∆Y ) is a Q-factorial dlt pair and Y is projective over S,

140 4. MINIMAL MODEL PROGRAM

(ii) KY + ∆Y is nef over S, and(iii) for any prime divisor E on X which is exceptional over Y , we

havea(E,X,∆) < a(E, Y,∆Y ).

Then (Y,∆Y ) is called a minimal model of (X,∆) over S. Furthermore,if KY +∆Y is semi-ample over S, then (Y,∆Y ) is called a good minimalmodel of (X,∆) over S.

Remark 4.8.6. By the same argument as Lemma 4.3.2, we canprove that a(E,X,∆) ≤ a(E, Y,∆Y ) for every prime divisor E over Xin Definition 4.8.5. Therefore, if (X,∆) is plt in Definition 4.8.5, then(Y,∆Y ) is plt and φ−1 has no exceptional divisors.

Definition 4.8.7 (Mori fiber spaces, see [Bir4, Definition 2.2]).Let (X,∆) be a log canonical pair and let π : X → S be a projectivemorphism onto a variety S. let φ : X 99K Y be a birational map over

S. We put ∆Y = ∆ + E where ∆ is the birational transform of ∆ onY and E is the reduced exceptional divisor of φ−1, that is, E =

∑j Ej

where Ej is a prime divisor on Y which is exceptional over X for everyj. We assume that

(i) (Y,∆Y ) is a Q-factorial dlt pair and Y is projective over S,(ii) there is a (KY +∆Y )-negative extremal contraction ϕ : Y → Z,

that is, −(KY +∆Y ) is ϕ-ample, ρ(Y/Z) = 1, and ϕ∗OY ' OZ ,over S with dimY > dimZ, and

(iii) we havea(E,X,∆) ≤ a(E, Y,∆Y ).

for any prime divisor E over X and strct inequality holds if Eis on X and φ contracts E.

Then (Y,∆Y ) is called a Mori fiber space of (X,∆) over S.

Let us quickly recall some results in [Bir4] and [HaX1]. For thedetails, see the original papers [Bir4] and [HaX1].

Theorem 4.8.8 (cf. [Bir4, Theorem 1.1] and [HaX1, Theorem1.6]). Let (X,∆) be a Q-factorial log canonical pair such that ∆ is aQ-divisor and let π : X → S be a projective morphism between quasi-projective varieties. Assume that there is an effective Q-divisor ∆′ onX such that (X,∆ + ∆′) is log canonical and KX + ∆ + ∆′ ∼Q,π 0.Then (X,∆) has a Mori fiber space or a good minimal model over S.

As a direct consequence of Theorem 4.8.8, we have:

Corollary 4.8.9. In Theorem 4.8.8, we further assume that KX+∆ is π-big. Then (X,∆) has a log canonical model over S.

4.8. MMP FOR LC PAIRS 141

Proof. Let (Y,∆Y ) be a good minimal model of (X,∆) over S.Then ⊕

m≥0

π∗OX(bm(KX + ∆)c) '⊕m≥0

πY ∗OY (bm(KY + ∆Y )c)

as OS-algebras (see Lemma 4.8.2), where πY : Y → S, and⊕m≥0

πY ∗OY (bm(KY + ∆Y )c)

is a finitely generated OS-algebra since KY + ∆Y is a πY -semi-ampleQ-divisor. We put

X ′ = ProjS⊕m≥0

π∗OX(bm(KX + ∆)c).

Then (X ′,∆′), where ∆′ is the strict transform of ∆ on X ′, is a logcanonical model of (X,∆) over S by Lemma 4.8.2.

Corollary 4.8.10 (cf. [Bir4, Corollary 1.2] and [HaX1, Corollary1.8]). Let ϕ : (X,∆) → W be a log canonical flipping contractionassociated to a (KX+∆)-negative extremal ray. Then the (KX+∆)-flipof ϕ : (X,∆)→ W exists.

Proof. Since ρ(X/W ) = 1, we may assume that ∆ is a Q-divisorby perturbing ∆ slightly. By taking an affine cover of W , we mayassume that W is affine. Then we can find an effective Q-divisor ∆′

on X such that KX + ∆ + ∆′ ∼Q,ϕ 0. Then (X,∆) has a log canonicalmodel over W . It is nothing but a flip of ϕ : (X,∆)→ W .

Remark 4.8.11. By Corollary 4.8.10, log canonical flips alwaysexist. On the other hand, log canonical flops do not always exist. Forthe details, see [F38, Section 7], where Kollar’s examples are describedin details.

4.8.12 (MMP for Q-factorial log canonical pairs). Let (X,∆) bea Q-factorial log canonical pair and let f : X → S be a projectivemorphism onto a variety S. By Theorem 4.5.2 and Corollary 4.8.10,we can run the minimal model program for (X,∆) over S. This meansthat the minimal model program discussed in 4.3.5 works by replacingdlt with log canonical. Moreover, by Theorem 4.5.2 (6), we can run theminimal model program with scaling discussed in 4.4.11 for Q-factoriallog canonical pairs. Note that the termination of the above minimalmodel programs is an important open problem of the minimal modeltheory (see Conjecture 4.3.6 and Lemma 4.9.3 below).

We note the following well-known lemma.

142 4. MINIMAL MODEL PROGRAM

Lemma 4.8.13 (see [KoMo, Corollary 3.44]). Let (X,∆) be a dlt(resp. klt or lc) pair. Let g : X 99K X ′ be either a divisorial contractionof a (KX + ∆)-negative extremal ray or a (KX + ∆)-flip. We put∆′ = g∗∆. Then (X ′,∆′) is also dlt (resp. klt or lc).

Proof. This lemma easily follows from Lemma 2.3.27 when (X,∆)is klt or lc. From now on, we treat the case when (X,∆) is dlt. LetZ ⊂ X be as in Proposition 2.3.20. We put Z ′ = g(Z) ∪ Exc(g−1)such that Exc(g−1) is the closed subset of X ′ where g−1 is not anisomorphism. Then X ′ \ Z ′ is isomorphic to an open subset of X \ Z.Therefore, X ′ \Z ′ is smooth and ∆′|X′\Z′ has a simple normal crossingsupport. Let E be an exceptional divisor over X ′ such that cX′(E), thecenter of E on X ′, is contained in Z ′. Then cX(E), the center of E onX, is contained in Z ∪ Exc(g), where Exc(g) is the closed subset of Xwhere g is not an isomorphism. We have

a(E,X ′,∆′) ≥ a(E,X,∆) ≥ −1

by Lemma 2.3.27. If cX(E) is contained in Z, then the second inequalityis strict by the definition of dlt pairs. If cX(E) is contained in Exc(g),then the first inequality is strict by Lemma 2.3.27. Anyway, (X ′,∆′)is dlt by Proposition 2.3.20.

We also note the following easy two propositions.

Proposition 4.8.14 (cf. [KoMo, Proposition 3.36]). Let (X,∆)be a Q-factorial log canonical pair and let π : X → S be a projectivemorphism. Let ϕR : X → Y be the contraction of a (KX + ∆)-negativeextremal ray R of NE(X/S). Assume that ϕR is either a divisorial ora Fano contraction. Then we have

(i) Y is Q-factorial, and(ii) ρ(Y/S) = ρ(X/S)− 1.

Proof. This proposition directly follows from Corollary 4.5.3 andCorollary 4.5.4.

Remark 4.8.15. If ϕR : X → Y is a Fano contraction in Proposi-tion 4.8.14, then we know that Y has only log canonical singularities by[F38]. We further assume that X has only log terminal singularities.Then Y has only log terminal singularities. For the details and somerelated topics, see [F38].

Proposition 4.8.16 (cf. [KoMo, Proposition 3.37]). Let (X,∆)be a Q-factorial log canonical pair and let π : X → S be a projective

4.8. MMP FOR LC PAIRS 143

morphism. Let ϕR : X → W be the flipping contraction of a (KX +∆)-negative extremal ray R of NE(X/S) and let ϕ+

R : X+ → W be the flip.

XϕR

AAA

AAAA

A

π

000

0000

0000

0000

φ //_______ X+

ϕ+R

zzzz

zzzz

W

S

Then we have

(i) X+ is Q-factorial, and(ii) ρ(X+/S) = ρ(X/S).

Proof. By perturbing ∆ slightly, we may assume that ∆ is a Q-divisor. Sine φ : X 99K X+ is an isomorphism in codimension one, itinduces a natural isomorphism between the group of Weil divisors onX and the group of Weil divisors on X+. Let D+ be a Weil divisor onX+ and let D be the strict transform of D+ on X. Then there is arational number r such that

(((D + r(KX + ∆)) ·R) = 0.

We take a positive integer m such that m(D + r(KX + ∆)) is Cartier.By Theorem 4.5.2 (4), there is a Cartier divisor DW on W such thatm(D + r(KX + ∆)) ∼ ϕ∗RDW . Thus we obtain that

mD+ = mφ∗D ∼ (ϕ+R)∗DW − (mr)(KX+ + ∆+)

is Q-Cartier. This means that X+ is Q-factorial. It is easy to see thatρ(X/S) = ρ(X+/S) by the above argument.

4.8.17 (Conjectures concerning MMP for lc pairs). The followingconjecture is one of the most important open problems of the minimalmodel program for log canonical pairs.

Conjecture 4.8.18. Let (X,∆) be a projective log canonical pairsuch that ∆ is a Q-divisor on X. Then the log canonical ring

R(X,∆) =⊕m≥0

H0(X,OX(bm(KX + ∆)c))

is a finitely generated C-algebra.

It is known that Conjecture 4.8.18 holds when dimX ≤ 4. WhendimX ≥ 5, Conjecture 4.8.18 is still an open problem.

144 4. MINIMAL MODEL PROGRAM

Theorem 4.8.19 (cf. [F22, Theorem 1.2]). Let (X,∆) be a pro-jective log canonical pair such that ∆ is a Q-divisor with dimX ≤ 4.Then the log canonical ring

R(X,∆) =⊕m≥0

H0(X,OX(bm(KX + ∆)c))

is a finitely generated C-algebra.

Let us recall the good minimal model conjecture.

Conjecture 4.8.20 (Good minimal model conjecture). Let (X,∆)be a Q-factorial projective dlt pair and let ∆ be an R-divisor. If KX+∆is pseudo-effective, then (X,∆) has a good minimal model.

In [FG2], we obtained:

Theorem 4.8.21. Conjecture 4.8.18 with dimX = n and Conjec-ture 4.8.20 with dimX ≤ n− 1 are equivalent.

Moreover, in [FG2], we proved:

Theorem 4.8.22. Conjecture 4.8.20 with dimX ≤ n − 1 is equiv-alent to Conjecture 4.8.23 with dimX = n.

Conjecture 4.8.23. Let (X,∆) be a Q-factorial projective plt pairsuch that ∆ is a Q-divisor on X and that b∆c is irreducible. We furtherassume that KX + ∆ is big. Then the log canonical ring

R(X,∆) =⊕m≥0

H0(X,OX(bm(KX + ∆)c))

is a finitely generated C-algebra.

Therefore, Conjecture 4.8.18 is equivalent to Conjecture 4.8.23 byTheorems 4.8.21 and 4.8.22.

Let us recall some related conjectures. For the details, see [FG1]and [FG2].

Conjecture 4.8.24 (Non-vanishing conjecture). Let X be a smoothprojective variety. If KX is pseudo-effective, then there exists some ef-fective Q-divisor D such that KX ∼Q D.

Remark 4.8.25. Let X be a smooth projective variety. Then KX

is pseudo-effective if and only if X is not uniruled by [BDPP]. For theproof, see, for example, [La2, Corollary 11.4.20].

Conjecture 4.8.26 (DLT extension conjecture, see [DHP, Con-jecture 1.3] and [FG2, Conjecture G]). Let (X,∆) be a projective

4.9. NON-Q-FACTORIAL MMP 145

dlt pair such that ∆ is a Q-divisor, b∆c = S, KX + ∆ is nef, andKX + ∆ ∼Q D ≥ 0 where S ⊂ SuppD. Then the restriction map

H0(X,OX(m(KX + ∆)))→ H0(S,OS(m(KX + ∆)))

is surjective for all sufficiently divisible integers m ≥ 2.

Note that the restriction map

H0(X,OX(m(KX + ∆)))→ H0(S,OS(m(KX + ∆)))

in Conjecture 4.8.26 is surjective for every positive integer m such thatm(KX + ∆) is Cartier when KX + ∆ is semi-ample (see [FG1, Propo-sition 5.12]). Therefore, Conjecture 4.8.26 follows from the abundanceconjecture (see Conjecture 4.7.5).

Conjecture 4.8.27. Let (X,∆) be a projective klt pair such that∆ is a Q-divisor with κ(X,KX + ∆) = 0. Then κσ(X,KX + ∆) = 0,where κσ denotes Nakayama’s numerical dimension in Definition 2.4.8.

Conjecture 4.8.27 is known as a special case of the generalized abun-dance conjecture for klt pairs.

In [FG2], we obtained the following two theorems.

Theorem 4.8.28. Conjecture 4.8.20 with dimX ≤ n follows fromConjecture 4.8.24 with dimX ≤ n and Conjecture 4.8.26 with dimX ≤n.

Theorem 4.8.29. Conjecture 4.8.20 with dimX ≤ n follows fromConjecture 4.8.24 with dimX ≤ n and Conjecture 4.8.27 with dimX ≤n.

Anyway, Conjecture 4.8.24 seems to be the hardest open problemin the minimal model program.

4.9. Non-Q-factorial MMP

In this section, we explain the minimal model program for non-Q-factorial log canonical pairs, that is, the minimal model program for(not necessarily Q-factorial) log canonical pairs. It is the most generalminimal model program in the usual sense. Although it is essentiallythe same as 4.3.5, we describe it for the reader’s convenience.

Let us explain the minimal model program for non-Q-factorial logcanonical pairs (cf. [F13, 4.4]).

4.9.1 (MMP for non-Q-factorial log canonical pairs). We start witha pair (X,∆) = (X0,∆0). Let f0 : X0 → S be a projective morphism.The aim is to set up a recursive procedure which creates intermediatepairs (Xi,∆i) and projective morphisms fi : Xi → S. After some steps,it should stop with a final pair (X ′,∆′) and f ′ : X ′ → S.

146 4. MINIMAL MODEL PROGRAM

Step 0 (Initial datum). Assume that we have already constructed(Xi,∆i) and fi : Xi → S with the following properties:

(1) (Xi,∆i) is log canonical,(2) fi is projective, and(3) Xi is not necessarily Q-factorial.

If Xi is Q-factorial, then it is easy to see that Xk is also Q-factorial forevery k ≥ i. Even when Xi is not Q-factorial, Xi+1 sometimes becomesQ-factorial (see, for example, Example 7.5.1 below.)

Step 1 (Preparation). If KXi+ ∆i is fi-nef, then we go directly to

Step 3 (2). If KXi+ ∆i is not fi-nef, then we have already established

the following two results (see Theorem 4.5.2):

(1) (Cone theorem). We have the following equality.

NE(Xi/S) = NE(Xi/S)(KXi+∆i)≥0 +

∑R≥0[Ci].

(2) (Contraction theorem). Any (KXi+ ∆i)-negative extremal

ray Ri ⊂ NE(Xi/S) can be contracted. Let ϕRi: Xi → Yi

denote the corresponding contraction. It sits in a commutativediagram.

Xi

ϕRi //

fi @@@

@@@@

@Yi

gi

S

Step 2 (Birational transformations). If ϕRi: Xi → Yi is birational,

then we take an effective Q-divisor ∆′i on Xi such that (Xi,∆′i) is log

canonical and −(KXi+ ∆′i) is ϕRi

-ample. Note that ρ(Xi/S) = 1. ByCorollary 4.8.9, the relative log canonical ring⊕

m≥0

(ϕRi)∗OXi

(bm(KXi+ ∆′i)c)

is a finitely generated OYi-algebra. We put

Xi+1 = ProjYi

⊕m≥0

(ϕRi)∗OXi

(bm(KXi+ ∆′i)c).

Then we have the following diagram.

Xi//_______

ϕRi @@@

@@@@

@Xi+1

ϕ+Ri

Yi

4.9. NON-Q-FACTORIAL MMP 147

Let ∆i+1 be the pushforward of ∆i on Xi+1. We note that (Xi+1,∆i+1)is the log canonical model of (Xi,∆i) over Yi (see Definition 4.8.1).Therefore, ϕ+

Ri: Xi+1 → Yi is a projective birational morphism, KXi+1

+

∆i+1 is ϕ+Ri

-ample, and (Xi+1,∆i+1) is log canonical. Then we go back

to Step 0 with (Xi+1,∆i+1), fi+1 = gi ϕ+Ri

and start anew.If Xi is Q-factorial, then so is Xi+1. If Xi is Q-factorial and ϕRi

is not small, then ϕ+Ri

: Xi+1 → Yi is an isomorphism. It may happenthat ρ(Xi/S) < ρ(Xi+1/S) whenXi is not Q-factorial (see, for example,Example 7.5.1 below).

Step 3 (Final outcome). We expect that eventually the procedurestops, and we get one of the following two possibilities:

(1) (Mori fiber space). If ϕRiis a Fano contraction, that is, dimYi <

dimXi, then we set (X ′,∆′) = (Xi,∆i) and f ′ = fi. We usu-ally call f ′ : (X ′,∆′) → Yi a Mori fiber space of (X,∆) overS.

(2) (Minimal model). If KXi+ ∆i is fi-nef, then we again set

(X ′,∆′) = (Xi,∆i) and f ′ = fi. We can easily check that(X ′,∆′) is a minimal model of (X,∆) over S in the sense ofDefinition 4.3.1.

We can always run the minimal model program discussed in 4.9.1for non-Q-factorial log canonical pairs. Unfortunately, the terminationof this minimal model program is widely open.

Remark 4.9.2. By Theorem 4.5.2 (6), we can run the minimalmodel program with scaling discussed in 4.4.11 for non-Q-factorial logcanonical pairs. Of course, the termination of this minimal modelprogram is an important open problem.

The following lemma is well known. It is essentially contained in[F13, Proof of Theorem 4.2.1].

Lemma 4.9.3. We assume that Conjecture 4.3.6 holds for Q-factorialdlt pairs in dimension n. Then the minimal model program discussedin 4.9.1 terminates after finitely many steps in dimension n.

Proof. Let

(X0,∆0) 99K (X1,∆1) 99K · · · 99K (Xk,∆k) 99K · · ·be a minimal model program discussed in 4.9.1 with dimX0 = n. Letα0 : X0

0 → X0 be a dlt blow-up, that is, (X00 ,∆

00) is Q-factorial and dlt

such that KX00

+ ∆00 = α∗0(KX0 + ∆0) (see Theorem 4.4.21). We run

the minimal model program with respet to KX00

+ ∆00 over Y0

X00 99K X1

0 99K X20 99K · · · ,

148 4. MINIMAL MODEL PROGRAM

and finally get a minimal model (Xk00 ,∆

k00 ) of (X0

0 ,∆00) over Y0. Since

(X1,∆1) → Y0 is the log canonical model of (X00 ,∆

00) → Y0, we have

a natural morphism α1 : Xk00 → X1 (see Lemma 4.8.3). We note that

KX

k00

+ ∆k00 = α∗1(KX1 + ∆1) by Lemma 4.8.3. We put (X0

1 ,∆01) =

(Xk00 ,∆

k00 ). We run the minimal model program with respect to KX0

1+

∆01 over Y1. Then we obtain a sequence

X01 99K X1

1 99K X21 99K · · · ,

and finally get a minimal model (Xk11 ,∆

k11 ) of (X0

1 ,∆01) over Y1. By

the same reason as above, we have a natural morphism α2 : Xk11 → X2

such that KX

k11

+ ∆k11 = α∗2(KX2 + ∆2) by Lemma 4.8.3. By repeating

this procedure, we obtain a (KX00

+ ∆00)-minimal model program over

S:X0

0 99K · · · 99K Xk00 = X0

1 99K · · · 99K Xk11 = X0

2 99K · · · .It terminates by the assumption of this lemma. Therefore, the originalminimal model program must terminate after finitely many steps.

By combining Lemma 4.9.3 with Lemma 4.3.8, it is sufficient toprove Conjecture 4.3.6 for klt pairs.

4.10. MMP for log surfaces

In this section, we discuss the minimal model theory for log surfaces,which is an application of Theorem 4.5.2.

Let us recall the definition of log surfaces.

Definition 4.10.1 (Log surfaces). Let X be a normal algebraicsurface and let ∆ be a boundary R-divisor on X such that KX + ∆ isR-Cartier. Then the pair (X,∆) is called a log surface. We recall thata boundary R-divisor is an effective R-divisor whose coefficients are lessthan or equal to one.

The following theorem is a special case of Theorem 4.5.2. Note thatthe non-lc locus of a log surface (X,∆) is zero-dimensional. Therefore,no curve is contained in the non-lc locus Nlc(X,∆) of (X,∆).

Theorem 4.10.2. Let (X,∆) be a log surface and let π : X → Sbe a projective morphism onto an algebraic variety S. Then we have

NE(X/S) = NE(X/S)KX+∆≥0 +∑

Rj

with the following properties.

(1) Rj is a (KX+∆)-negative extremal ray of NE(X/S) for everyj.

4.10. MMP FOR LOG SURFACES 149

(2) Let H be a π-ample R-divisor on X. Then there are onlyfinitely many Rj’s included in (KX +∆+H)<0. In particular,the Rj’s are discrete in the half-space (KX + ∆)<0.

(3) Let R be a (KX+∆)-negative extremal ray of NE(X/S). Thenthere exists a contraction morphism ϕR : X → Y over S withthe following properties.(i) Let C be an integral curve on X such that π(C) is a point.

Then ϕR(C) is a point if and only if [C] ∈ R.(ii) OY ' (ϕR)∗OX .(iii) Let L be a line bundle on X such that L ·C = 0 for every

curve C with [C] ∈ R. Then there exists a line bundle LYon Y such that L ' ϕ∗RLY .

Theorem 4.10.3 and Theorem 4.10.4 are the main results of [F29].

Theorem 4.10.3 (Minimal model program for log surfaces ([F29,Theorem 3.3])). Let (X,∆) be a log surface and let π : X → S be aprojective morphism onto an algebraic variety S. We assume one ofthe following conditions:

(A) X is Q-factorial.(B) (X,∆) is log canonical.

Then, by Theorem 4.10.2, we can run the minimal model program overS with respect to KX+∆. So, there is a sequence of at most ρ(X/S)−1contractions

(X,∆) = (X0,∆0)ϕ0−→ (X1,∆1)

ϕ1−→ · · · ϕk−1−→ (Xk,∆k) = (X∗,∆∗)

over S such that one of the following holds:

(1) (Minimal model). if KX + ∆ is pseudo-effective over S, thenKX∗ + ∆∗ is nef over S. In this case, (X∗,∆∗) is called aminimal model of (X,∆) over S.

(2) (Mori fiber space). if KX + ∆ is not pseudo-effective overS, then there is a morphism g : X∗ → C over S such that−(KX∗ + ∆∗) is g-ample, dimC < 2, and ρ(X∗/C) = 1. Weusually call g : (X∗,∆∗) → C a Mori fiber space of (X,∆)over S.

We note that Xi is Q-factorial (resp. (Xi,∆i) is log canonical) for everyi in Case (A) (resp. (B)).

Theorem 4.10.4 (Abundance theorem ([F29, Theorem 8.1])). Let(X,∆) be a log surface and let π : X → S be a proper surjectivemorphism onto a variety S. Assume that X is Q-factorial or that(X,∆) is log canonical. We further assume that KX + ∆ is π-nef.Then KX + ∆ is π-semi-ample.

150 4. MINIMAL MODEL PROGRAM

As an easy consequence of Theorem 4.10.3 and Theorem 4.10.4, wehave:

Theorem 4.10.5. Let X be a normal Q-factorial projective surface.Then the canonical ring

R(X) =⊕m≥0

H0(X,OX(mKX))

is a finitely generated C-algebra.

As a corollary of Theorem 4.10.5, we obtain:

Theorem 4.10.6 ([F29, Corollary 4.6]). Let X be a normal pro-jective surface with only rational singularities. Then the canonical ring

R(X) =⊕m≥0

H0(X,OX(mKX))

is a finitely generated C-algebra.

Remark 4.10.7. If X is a surface with only rational singularities,then it is well known that X is Q-factorial. If X has only rationalsingularities in Theorem 4.10.3, then we can check that Xi has onlyrational singularities for every i (see [F29, Proposition 3.7]).

The following theorem, which is not covered by Theorem 4.10.5, iswell known to the experts (see, for example, [Ba, Theorem 14.42]).

Theorem 4.10.8. Let X be a normal projective Gorenstein surface.Then the canonical ring

R(X) =⊕m≥0

H0(X,OX(mKX))

is a finitely generated C-algebra.

Here, we give a proof of Theorem 4.10.8 by using Theorem 4.5.2.

Proof. If κ(X,KX) ≤ 0, then the statement is obvious. Therefore,we may assume κ(X,KX) ≥ 1. By taking some crepant resolutions ofrational Gorenstein singularities, we may assume that every singularityof X is not log terminal. Let f : Y → X be a minimal resolution ofsingularities of X. Then we can write KY + E = f ∗KX where E is aneffective Cartier divisor with Exc(f) = SuppE by the negativity lemma(see Lemma 2.3.26). We assume that KX is not nef. Then KY + E isobviously not nef. By Theorem 4.5.2, there is an irreducible rationalcurve C ′ on Y such that C ′ · (KY + E) < 0 and (C ′)2 < 0. Note thatf(C ′) = C is not a point by C ′ · (KY + E) < 0. Therefore, C ′ · E ≥ 0.This implies C ′ · KY < 0. Thus, we have (C ′)2 = C ′ · KY = −1.

4.10. MMP FOR LOG SURFACES 151

Note that E is an effective Cartier divisor. So, we have C ′ · E = 0 byC ′ ·(KY +E) < 0 and C ′ ·KY = −1. This implies that C ′∩E = ∅. ThusC is contained in the smooth locus of X. Note that C ⊂ B(KX) ( X.Let ϕ : X → X ′ be the contraction morphism which contracts C to asmooth point. We can replace X with X ′. By repeating this processfinitely many times, we may assume that KX is nef. Note that R(X) ispreserved by this process. If κ(X,KX) = 2, then KX is semi-ample bythe basepoint-free theorem (see Corollary 4.5.6). Note that the non-kltlocus of X is zero-dimensional. If κ(X,KX) = 1, then it is easy tosee that KX is semi-ample. Anyway, we obtain that the canonical ringR(X) is a finitely generated C-algebra.

We do not know if Theorem 4.10.8 holds true or not under theweaker assumption that X is only Q-Gorenstein. The following theo-rem is a partial result for Q-Gorenstein surfaces.

Theorem 4.10.9. Let X be a normal projective surface such thatKX is Q-Cartier. Assume that there exists an effective Q-divisor D =∑

i diDi such that KX ∼Q D and that Di is a Q-Cartier prime divisoron X for every i. Then the canonical ring

R(X) =⊕m≥0

H0(X,OX(mKX))

is a finitely generated C-algebra.

Proof. Let R = R≥0[C] be aKX-negative extremal ray ofNE(X).Then C · KX < 0 implies that C ⊂ SuppD. So, NE(X) has onlyfinitely many KX-negative extremal rays. Moreover, the contractionmorphism ϕR : X → Y is birational. We note that the exceptionallocus of ϕR is an irreducible curve contained in SuppD. This is becauseeach irreducible component of D is Q-Cartier. Therefore, we can checkthat KY = ϕR∗KX is Q-Cartier, KY ∼Q

∑i diϕR∗Di, and ϕR∗Di is Q-

Cartier if ϕR∗Di 6= 0. After finitely many contraction morphisms, thisprogram terminates. Since R(X) is preserved by the above process, wemay assume that KX is nef by replacing X with its final model. Whenκ(X,KX) = 0 or 1, R(X) is obviously a finitely generated C-algebra.So, we may assume that KX is big. Since the non-klt locus of Xis zero-dimensional, KX is semi-ample by the basepoint-free theorem(see Corollary 4.5.6). In particular, R(X) is a finitely generated C-algebra.

For the details of the minimal model theory for log surfaces, see[F29]. For the minimal model theory for log surfaces in positive char-acteristic, see [FT], [Tana1], and [Tana2]. In positive characteristic,

152 4. MINIMAL MODEL PROGRAM

we note the following contraction theorem of Artin–Keel (see [Ar] and[Ke]).

Theorem 4.10.10 (Artin–Keel). Let X be a complete normal al-gebraic surface defined over an algebraically closed field k of positivecharacteristic and let H be a nef and big Cartier divisor on X. We put

E(H) = C |C is a curve on X and C ·H = 0.

Then E(H) consists of finitely many irreducible curves on X. Assumethat H|E(H) is semi-ample where

E(H) =∪

C∈E(H)

C

with the reduced scheme structure. Then H is semi-ample. Therefore,

Φ|mH| : X → Y

is a proper birational morphism onto a normal projective surface Ywhich contracts E(H) and is an isomorphism outside E(H) for a suf-ficiently large and divisible positive integer m.

For the proof of Theorem 4.10.10, we recommend the reader to see[FT, Theorem 2.1], where we gave two different proofs using the Fujitavanishing theorem (see Theorem 3.8.1). Note that Theorem 4.10.10does not hold in characteristic zero. By Theorem 4.10.10, the minimalmodel theory for log surfaces is easier in characteristic p > 0 than incharacteristic zero.

We close this section with an example of a non-Q-factorial logcanonical surface.

Example 4.10.11 (Non-Q-factorial log canonical surface). Let C ⊂P2 be a smooth cubic curve and let Y ⊂ P3 be a cone over C. ThenY is log canonical. In this case, Y is not Q-factorial. We can check itas follows. Let f : X = PC(OC ⊕ L) → Y be a resolution such thatKX +E = f ∗KY , where L = OP2(1)|C and E is the exceptional curve.We take P,Q ∈ C such that OC(P − Q) is not a torsion in Pic0(C).We consider D = π∗P − π∗Q, where π : X = PC(OC ⊕ L) → C. Weput D′ = f∗D. If D′ is Q-Cartier, then mD = f ∗mD′ + aE for somea ∈ Z and m ∈ Z>0. Restrict it to E. Then

OC(m(P −Q)) ' OE(aE) ' (L−1)⊗a.

Therefore, we obtain that a = 0 and m(P −Q) ∼ 0. This is a contra-diction. Thus, D′ is not Q-Cartier. In particular, Y is not Q-factorial.

4.11. ON SEMI LOG CANONICAL PAIRS 153

4.11. On semi log canonical pairs

In this final section, we quickly review the main theorem of [F33],which says that every quasi-projective semi log canonical pair has a nat-ural quasi-log structure compatible with the original semi log canonicalstructure, without proof. For the details, see [F33].

The notion of semi log canonical singularities was introduced in[KSB] in order to investigate deformations of surface singularities andcompactifications of moduli spaces for surfaces of general type. By therecent developments of the minimal model program, we know that theappropriate singularities to permit on the varieties at the boundaries ofmoduli spaces are semi log canonical (see, for example, [Ale1], [Ale2],[Ko13], [HaKo, Part III], [Kv4], [Kv7], [KSB], and so on).

First, let us recall the definition of conductors.

Definition 4.11.1 (Conductor). LetX be an equidimensional vari-ety which satisfies Serre’s S2 condition and is normal crossing in codi-mension one and let ν : Xν → X be the normalization. Then theconductor ideal of X is defined by

condX := HomOX(ν∗OXν ,OX) ⊂ OX .

The conductor CX of X is the subscheme defined by condX . Since Xsatisfies Serre’s S2 condition and X is normal crossing in codimensionone, CX is a reduced closed subscheme of pure codimension one in X.

Although we do not use the notion of double normal crossing pointsand pinch points explicitly in this book, it plays crucial roles for thestudy of semi log canonical pairs.

Definition 4.11.2 (Double normal crossing points and pinch points).An n-dimensional singularity (x ∈ X) is called a double normal cross-ing point if it is analytically (or formally) isomorphic to

(0 ∈ (x0x1 = 0)) ⊂ (0 ∈ Cn+1).

It is called a pinch point if it is analytically (or formally) isomorphic to

(0 ∈ (x20 = x1x

22)) ⊂ (0 ∈ Cn+1).

We recall the definition of semi log canonical pairs.

Definition 4.11.3 (Semi log canonical pairs). Let X be an equidi-mensional algebraic variety that satisfies Serre’s S2 condition and isnormal crossing in codimension one. Let ∆ be an effective R-divisorwhose support does not contain any irreducible components of the con-ductor of X. The pair (X,∆) is called a semi log canonical pair (an slcpair, for short) if

154 4. MINIMAL MODEL PROGRAM

(1) KX + ∆ is R-Cartier, and(2) (Xν ,Θ) is log canonical, where ν : Xν → X is the normaliza-

tion and KXν + Θ = ν∗(KX + ∆).

Note that if X has only smooth points, double normal crossingpoints and pinch points then it is easy to see that X is semi log canon-ical.

The following examples are obvious by the definition of semi logcanonical pairs.

Example 4.11.4. Let (X,∆) be a log canonical pair. Then (X,∆)is a semi log canonical pair.

Example 4.11.5. Let (X,∆) be a semi log canonical pair. Assumethat X is normal. Then (X,∆) is log canonical.

Example 4.11.6. Let X be a nodal curve. More generally, X is anormal crossing variety. Then X is semi log canonical.

We introduce the notion of semi log canonical centers. It is a directgeneralization of the notion of log canonical centers for log canonicalpairs.

Definition 4.11.7 (Slc center). Let (X,∆) be a semi log canonicalpair and let ν : Xν → X be the normalization. We set

KXν + Θ = ν∗(KX + ∆).

A closed subvariety W of X is called a semi log canonical center (anslc center, for short) with respect to (X,∆) if there exist a resolutionof singularities f : Y → Xν and a prime divisor E on Y such that thediscrepancy coefficient a(E,Xν ,Θ) = −1 and ν f(E) = W .

For our purposes, it is very convenient to introduce the notion ofslc strata for semi log canonical pairs.

Definition 4.11.8 (Slc stratum). Let (X,∆) be a semi log canoni-cal pair. A subvariety W of X is called a semi log canonical stratum (anslc stratum, for short) of the pair (X,∆) if W is a semi log canonicalcenter with respect to (X,∆) or W is an irreducible component of X.

The following theorem is the main theorem of [F33].

Theorem 4.11.9 ([F33, Theorem 1.2]). Let (X,∆) be a quasi-projective semi log canonical pair. Then we can construct a smoothquasi-projective variety M with dimM = dimX + 1, a simple normalcrossing divisor Z on M , a subboundary R-divisor B on M , and a pro-jective surjective morphism h : Z → X with the following properties.

4.11. ON SEMI LOG CANONICAL PAIRS 155

(i) B and Z have no common irreducible components.(ii) Supp(Z +B) is a simple normal crossing divisor on M .(iii) KZ + ∆Z ∼R h

∗(KX + ∆) with ∆Z = B|Z.(iv) h∗OZ(d−∆<1

Z e) ' OX .

By properties (i), (ii), (iii), and (iv), [X,KX +∆] has a quasi-log struc-ture with only qlc singularities.

(v) The set of slc strata of (X,∆) gives the set of qlc strata of[X,KX + ∆]. This means that W is an slc stratum of (X,∆)if and only if W is the h-image of some stratum of the simplenormal crossing pair (Z,∆Z).

By property (v), the above quasi-log structure of [X,KX + ∆] is com-patible with the original semi log canonical structure of (X,∆).

We note that h∗OZ ' OX by condition (iv).

For the details of quasi-log structures, see Chapter 6.

Remark 4.11.10. In Theorem 4.11.9, h : Z → X is not necessar-ily birational. Note that Z is not always irreducible even when X isirreducible.

Example 4.11.11. Let us consider the Whitney umbrella

X = (x2 − y2z = 0) ⊂ C3.

It is easy to see that X has only semi log canonical singularities. LetM → C3 be the blow-up along C = (x = y = 0). We put Z =X ′ + E, where X ′ is the strict transform of X on M and E is theexceptional divisor of M → C3. Then Z is a simple normal crossingvariety on a smooth quasi-projective variety M . Let h : Z → X bethe natural morphism. Then we can easily check that h∗OZ ' OXand KZ = h∗KX . In this case, h : Z → X gives a natural quasi-log structure which is compatible with the original semi log canonicalstructure. Note that h : Z → X is not birational.

By Theorem 4.11.9, we can apply the fundamental theorems forquasi-log schemes (see Chapter 6) to quasi-projective semi log canonicalpairs. For the details, see [F33]. Moreover, Theorem 4.11.9 drasticallyincreased the importance of the theory of quasi-log schemes.

We note that the proof of Theorem 4.11.9 heavily depends on therecent developments of the theory of partial resolution of singularitiesfor reducible varieties (see [BM] and [BVP]).

We close this section with the definition of stable varieties.

Definition 4.11.12 (Stable varieties). Let X be a projective vari-ety with only semi log canonical singularities. If KX is ample, then Xis called a stable variety.

156 4. MINIMAL MODEL PROGRAM

Definition 4.11.12 is a generalization of the notion of stable curves.Roughly speaking, the notion of semi log canonical singularities wasintroduced in [KSB] in order to define stable surfaces for the compact-ification problem of moduli spaces of canonically polarized surfaces.Although the approach to the moduli problems in [KSB] is not di-rectly related to Mumford’s geometric invariant theory, the notion ofsemi log canonical singularities appears to be natural from the geomet-ric invariant theoretic viewpoint by [Od].

CHAPTER 5

Injectivity and vanishing theorems

The main purpose of this chapter is to establish the vanishingand torsion-free theorem for simple normal crossing pair (see Theo-rem 5.1.3), which is indispensable for the theory of quasi-log schemesdiscussed in Chapter 6.

In Section 5.1, we explain the main results of this chapter. In Sec-tion 5.2, we review the notion of Q-divisors and R-divisors again forthe reader’s convenience. This is because we have to treat reduciblevarieties from this chapter. In Section 5.3, we quickly review Du Boiscomplexes and Du Bois singularities. We use them in Section 5.4. InSection 5.4, we prove the Hodge theoretic injectivity theorem. It is acorrect and powerful generalization of Kollar’s injectivity theorem fromthe Hodge theoretic viewpoint. In Section 5.5, we generalize the Hodgetheoretic injectivity theorem for the relative setting. The relative ver-sion of the Hodge theoretic injectivity theorem drastically simplifiesthe proof of the injectivity, vanishing, and torsion-free theorems forsimple normal crossing pairs in Section 5.6. Section 5.6 is devoted tothe proof of the injectivity, vanishing, and torsion-free theorems forsimple normal crossing pairs. In Section 5.7, we treat the vanishingtheorem of Reid–Fukuda type for embedded simple normal crossingpairs. In Section 5.8, we treat embedded normal crossing pairs. Notethat the results in Section 5.8 are not necessary for the theory of quasi-log schemes discussed in Chapter 6. So the reader can skip Section 5.8.Section 5.9 contains many nontrivial examples, which help us under-stand the results discussed in this chapter.

5.1. Main results

In this chapter, we prove the following theorems. Theorem 5.1.1 isa complete generalization of Lemma 3.1.1.

Theorem 5.1.1 (Hodge theoretic injectivity theorem, see Theorem5.4.1). Let (X,∆) be a simple normal crossing pair such that X isproper and that ∆ is a boundary R-divisor on X. Let L be a Cartierdivisor on X and let D be an effective Weil divisor on X whose supportis contained in Supp ∆. Assume that L ∼R KX + ∆. Then the natural

157

158 5. INJECTIVITY AND VANISHING THEOREMS

homomorphism

Hq(X,OX(L))→ Hq(X,OX(L+D))

induced by the inclusion OX → OX(D) is injective for every q.

After we generalize Theorem 5.1.1 for the relative setting, we proveTheorem 5.1.2 as an application. It is a generalization of Kollar’s in-jectivity theorem: Theorem 3.6.2.

Theorem 5.1.2 (Injectivity theorem for simple normal crossingpairs, see Theorem 5.6.2). Let (X,∆) be a simple normal crossing pairsuch that ∆ is a boundary R-divisor on X, and let π : X → V be aproper morphism between schemes. Let L be a Cartier divisor on Xand let D be an effective Cartier divisor that is permissible with respectto (X,∆). Assume the following conditions.

(i) L ∼R,π KX + ∆ +H,(ii) H is a π-semi-ample R-divisor, and(iii) tH ∼R,π D+D′ for some positive real number t, where D′ is an

effective R-Cartier R-divisor that is permissible with respect to(X,∆).

Then the homomorphisms

Rqπ∗OX(L)→ Rqπ∗OX(L+D),

which are induced by the natural inclusion OX → OX(D), are injectivefor all q.

By using Theorem 5.1.2, we obtain Theorem 5.1.3.

Theorem 5.1.3 (see Theorem 5.6.3). Let (Y,∆) be a simple normalcrossing pair such that ∆ is a boundary R-divisor on Y . Let f : Y → Xbe a proper morphism to a scheme X and let L be a Cartier divisoron Y such that L − (KY + ∆) is f -semi-ample. Let q be an arbitrarynon-negative integer. Then we have the following properties.

(i) Every associated prime of Rqf∗OY (L) is the generic point ofthe f -image of some stratum of (Y,∆).

(ii) Let π : X → V be a projective morphism to a scheme V suchthat

L− (KY + ∆) ∼R f∗H

for some π-ample R-divisor H on X. Then Rqf∗OY (L) isπ∗-acyclic, that is,

Rpπ∗Rqf∗OY (L) = 0

for every p > 0.

5.1. MAIN RESULTS 159

Theorem 5.1.3 will play crucial roles in the theory of quasi-logschemes discussed in Chapter 6.

We give an easy example, which shows a trouble in [Am1].

Example 5.1.4. Let X be a smooth projective variety and let H bea Cartier divisor on X. Let A be a smooth irreducible member of |2H|and let S be a smooth divisor on X such that S and A are disjoint.We put B = 1

2A + S and L = H +KX + S. Then L ∼Q KX + B and

2L ∼ 2(KX +B). We define

E = OX(−L+KX)

as in the proof of [Am1, Theorem 3.1]. Apply the argument in theproof of [Am1, Theorem 3.1]. Then we have a double cover π : Y → Xcorresponding to 2B ∈ |E−2|. Then

π∗ΩpY (log π∗B) ' Ωp

X(logB)⊕ ΩpX(logB)⊗ E(S).

Note that ΩpX(logB) ⊗ E is not a direct summand of π∗Ω

pY (log π∗B).

Theorem 3.1 in [Am1] claims that the homomorphisms

Hq(X,OX(L))→ Hq(X,OX(L+D))

are injective for all q. Here, we used the notation in [Am1, Theorem3.1]. In our case, D = mA for some positive integer m. However,Ambro’s argument just implies that

Hq(X,OX(L− bBc))→ Hq(X,OX(L− bBc+D))

is injective for every q. Therefore, his proof works only for the casewhen bBc = 0 even if X is smooth.

The proof of [Am1, Theorem 3.1] seems to contain a conceptualmistake. The trouble discussed in Example 5.1.4 is serious for ap-plications to the theory of quasi-log schemes. Ambro’s proof of theinjectivity theorem in [Am1] is based on the mixed Hodge structure of

H i(Y − π∗B,Z).

It is a standard technique for injectivity and vanishing theorems inthe minimal model program. In this chapter, we use the mixed Hodgestructure of

H ic(Y − π∗S,Z),

where H ic(Y − π∗S,Z) is the cohomology group with compact support.

5.1.5 (Observation). Let us explain the main idea of this chapter.Let X be a smooth projective variety with dimX = n and let ∆ be a

160 5. INJECTIVITY AND VANISHING THEOREMS

simple normal crossing divisor on X. The decomposition

H ic(X −∆,C) =

⊕p+q=i

Hq(X,ΩpX(log ∆)⊗OX(−∆)).

is suitable for our purposes. The dual statement

H2n−i(X −∆,C) =⊕p+q=i

Hn−q(X,Ωn−pX (log ∆)),

which is well known and is commonly used is not useful for our pur-poses. Note that the paper [FF] supports our approach in this chapter.

Anyway, [Am1, 3. Vanishing theorems] seems to be quite short.In this chapter, we establish the injectivity, vanishing, and torsion-free theorems sufficient for the theory of quasi-log schemes discussedin Chapter 6. This chapter covers all the results in [Am1, Section3] and contains several nontrivial generalizations. In [Am1, Section3], Ambro closely followed Esnault–Viehweg’s arguments in [EsVi2](see also [F17, Chapter 2]). On the other hand, our approach in thischapter is more similar to Kollar’s (see, for example, [Ko6, Chapter 9]and [KoMo, Section 2.4]).

5.2. Simple normal crossing pairs

We quickly recall basic definitions of divisors again. We note thatwe have to deal with reducible schemes in this paper. For details, see,for example, [Har5, Section 2] and [Li, Section 7.1].

5.2.1. Let X be a scheme with structure sheaf OX and let KX bethe sheaf of total quotient rings of OX . Let K∗X denote the (multiplica-tive) sheaf of invertible elements in KX , and O∗X the sheaf of invertibleelements in OX . We note that OX ⊂ KX and O∗X ⊂ K∗X .

5.2.2 (Cartier, Q-Cartier, and R-Cartier divisors). A Cartier divisorD on X is a global section of K∗X/O∗X , that is, D is an element ofH0(X,K∗X/O∗X). A Q-Cartier divisor (resp. R-Cartier divisor) is anelement of H0(X,K∗X/O∗X)⊗Z Q (resp. H0(X,K∗X/O∗X)⊗Z R).

5.2.3 (Linear, Q-linear, and R-linear equivalence). Let D1 and D2

be two R-Cartier divisors on X. Then D1 is linearly (resp. Q-linearly,or R-linearly) equivalent to D2, denoted by D1 ∼ D2 (resp. D1 ∼Q D2,or D1 ∼R D2) if

D1 = D2 +k∑i=1

ri(fi)

5.2. SIMPLE NORMAL CROSSING PAIRS 161

such that fi ∈ Γ(X,K∗X) and ri ∈ Z (resp. ri ∈ Q, or ri ∈ R) for everyi. We note that (fi) is a principal Cartier divisor associated to fi, thatis, the image of fi by

Γ(X,K∗X)→ Γ(X,K∗X/O∗X).

Let f : X → Y be a morphism. If there is an R-Cartier divisor B on Ysuch that D1 ∼R D2 + f ∗B, then D1 is said to be relatively R-linearlyequivalent to D2. It is denoted by D1 ∼R,f D2 or D1 ∼R,Y D2.

5.2.4 (Supports). Let D be a Cartier divisor on X. The support ofD, denoted by SuppD, is the subset of X consisting of points x suchthat a local equation for D is not in O∗X,x. The support of D is a closedsubset of X.

5.2.5 (Weil divisors, Q-divisors, and R-divisors). LetX be an equidi-mensional variety. We note that X is not necessarily regular in codi-mension one. A (Weil) divisor D on X is a finite formal sum

n∑i=1

diDi

where Di is an irreducible reduced closed subscheme of X of pure codi-mension one and di is an integer for every i such that Di 6= Dj fori 6= j.

If di ∈ Q (resp. di ∈ R) for every i, then D is called a Q-divisor(resp. R-divisor). We define the round-up dDe =

∑ri=1ddieDi (resp. the

round-down bDc =∑r

i=1bdicDi), where for every real number x, dxe(resp. bxc) is the integer defined by x ≤ dxe < x+1 (resp. x−1 < bxc ≤x). The fractional part D of D denotes D − bDc. We define D<1 =∑

di<1 diDi, and so on. We call D a boundary (resp. subboundary) R-divisor if 0 ≤ di ≤ 1 (resp. di ≤ 1) for every i.

Let us define simple normal crossing pairs.

Definition 5.2.6 (Simple normal crossing pairs). We say that thepair (X,D) is simple normal crossing at a point a ∈ X ifX has a Zariskiopen neighborhood U of a that can be embedded in a smooth varietyY , where Y has regular system of parameters (x1, · · · , xp, y1, · · · , yr)at a = 0 in which U is defined by a monomial equation

x1 · · ·xp = 0

and

D =r∑i=1

αi(yi = 0)|U , αi ∈ R.

162 5. INJECTIVITY AND VANISHING THEOREMS

We say that (X,D) is a simple normal crossing pair if it is simple nor-mal crossing at every point of X. If (X, 0) is a simple normal crossingpair, then X is called a simple normal crossing variety. If X is a sim-ple normal crossing variety, then X has only Gorenstein singularities.Thus, it has an invertible dualizing sheaf ωX . Therefore, we can definethe canonical divisor KX such that ωX ' OX(KX) (cf. [Li, Section 7.1Corollary 1.19]). It is a Cartier divisor on X and is well-defined up tolinear equivalence.

We say that a simple normal crossing pair is embedded if thereexists a closed embedding ι : X →M , where M is a smooth variety ofdimension dimX + 1. We call M the ambient space of (X,∆).

The author learned the following interesting example from KentoFujita (cf. [Ko13, Remark 1.9]).

Example 5.2.7. Let X1 = P2 and let C1 be a line on X1. LetX2 = P2 and let C2 be a smooth conic on X2. We fix an isomorphismτ : C1 → C2. By gluing X1 and X2 along τ : C1 → C2, we obtain asimple normal crossing surface X such that −KX is ample (cf. [Fk1]).We can check that X can not be embedded into any smooth varietiesas a simple normal crossing divisor.

We note that a simple normal crossing pair is called a semi-snc pairin [Ko13, Definition 1.9].

Definition 5.2.8 (Strata and permissibility). Let X be a simplenormal crossing variety and let X =

∪i∈I Xi be the irreducible de-

composition of X. A stratum of X is an irreducible component ofXi1 ∩ · · · ∩Xik for some i1, · · · , ik ⊂ I. A Cartier divisor D on X ispermissible if D contains no strata of X in its support. A finite Q-linear(resp. R-linear) combination of permissible Cartier divisors is called apermissible Q-divisor (resp. R-divisor) on X.

5.2.9. Let X be a simple normal crossing variety. Let PerDiv(X)be the abelian group generated by permissible Cartier divisors on Xand let Weil(X) be the abelian group generated by Weil divisors on X.Then we can define natural injective homomorphisms of abelian groups

ψ : PerDiv(X)⊗Z K→Weil(X)⊗Z K

5.2. SIMPLE NORMAL CROSSING PAIRS 163

for K = Z, Q, and R. Let ν : X → X be the normalization. Then wehave the following commutative diagram.

Div(X)⊗Z K ∼eψ

// Weil(X)⊗Z Kν∗

PerDiv(X)⊗Z K

ψ//

ν∗

OO

Weil(X)⊗Z K

Note that Div(X) is the abelian group generated by Cartier divisors

on X and that ψ is an isomorphism since X is smooth.By ψ, every permissible Cartier divisor (resp. Q-divisor or R-divisor)

can be considered as a Weil divisor (resp. Q-divisor or R-divisor). ForQ-divisors and R-divisors, see 5.2.5. Therefore, various operations, forexample, bDc, D<1, and so on, make sense for a permissible R-divisorD on X.

We note the following easy example.

Example 5.2.10. Let X be a simple normal crossing variety inC3 = Spec C[x, y, z] defined by xy = 0. We set D1 = (x + z = 0) ∩Xand D2 = (x − z = 0) ∩ X. Then D = 1

2D1 + 1

2D2 is a permissible

Q-divisor on X. In this case, bDc = (x = z = 0) on X. Therefore,bDc is not a Cartier divisor on X.

Definition 5.2.11 (Simple normal crossing divisors). Let X be asimple normal crossing variety and let D be a Cartier divisor on X. If(X,D) is a simple normal crossing pair and D is reduced, then D iscalled a simple normal crossing divisor on X.

Remark 5.2.12. Let X be a simple normal crossing variety andlet D be a K-divisor on X where K = Q or R. If SuppD is a simplenormal crossing divisor on X and D is K-Cartier, then bDc and dDe(resp. D, D<1, and so on) are Cartier (resp. K-Cartier) divisors onX (cf. [BVP, Section 8]).

The following lemma is easy but important.

Lemma 5.2.13. Let X be a simple normal crossing variety and let Bbe a permissible R-divisor on X such that bBc = 0. Let A be a Cartierdivisor on X. Assume that A ∼R B. Then there exists a permissibleQ-divisor C on X such that A ∼Q C, bCc = 0, and SuppC = SuppB.

Proof. We can write B = A +∑k

i=1 ri(fi), where fi ∈ Γ(X,K∗X)and ri ∈ R for every i. Here, KX is the sheaf of total quotient rings of

164 5. INJECTIVITY AND VANISHING THEOREMS

OX (see 5.2.1). Let P ∈ X be a scheme theoretic point correspondingto some stratum of X. We consider the following affine map

Kk → H0(XP ,K∗XP/O∗XP

)⊗Z K

given by (a1, · · · , ak) 7→ A +∑k

i=1 ai(fi), where XP = SpecOX,P andK = Q or R. Then we can check that

P = (a1, · · · , ak) ∈ Rk |A+∑i

ai(fi) is permissible ⊂ Rk

is an affine subspace of Rk defined over Q. Therefore, we see that

S = (a1, · · · , ak) ∈ P | Supp(A+∑i

ai(fi)) ⊂ SuppB ⊂ P

is an affine subspace of Rk defined over Q. Since (r1, · · · , rk) ∈ S, weknow that S 6= ∅. We take a point (s1, · · · , sk) ∈ S ∩ Qk which isgeneral in S and sufficiently close to (r1, · · · , rk) and set

C = A+k∑i=1

si(fi).

By construction, C is a permissible Q-divisor such that C ∼Q A, bCc =0, and SuppC = SuppB.

We need the following important definition in Section 5.6.

Definition 5.2.14 (Strata and permissibility for pairs). Let (X,D)be a simple normal crossing pair. Let ν : Xν → X be the normalization.We define Θ by the formula

KXν + Θ = ν∗(KX +D).

Then a stratum of (X,D) is an irreducible component of X or the ν-image of a log canonical center of (Xν ,Θ). When D = 0, this definitionis compatible with Definition 5.2.8. A Cartier divisor B on X is per-missible with respect to (X,D) if B contains no strata of (X,D) in itssupport. A finite Q-linear (resp. R-linear) combination of permissibleCartier divisors with respect to (X,D) is called a permissible Q-divisor(resp. R-divisor) with respect to (X,D).

5.2.15 (Partial resolution of singularities for reducible varieties). Inthis chapter, we will repeatedly use the following results on the partialresolution of singularities for reducible varieties.

Theorem 5.2.16 is a special case of [BM, Theorem 1.5].

5.3. DU BOIS COMPLEXES AND DU BOIS PAIRS 165

Theorem 5.2.16 (Bierstone–Milman). Let X be an equidimen-sional variety and let Xsnc denote the simple normal crossings locusof X. Then there is a morphism σ : X ′ → X which is a composite offinitely many blow-ups such that

(1) X ′ is a simple normal crossing variety,(2) σ is an isomorphism over Xsnc, and(3) σ maps SingX ′ birationally onto the closure of SingXsnc.

Theorem 5.2.17 is a special case of [BVP, Theorem 1.4].

Theorem 5.2.17 (Bierstone–Vera Pacheco). Let X be an equidi-mensional variety and let ∆ be an R-divisor on X. Assume that nocomponent of ∆ lies in the singular locus of X. Let U ⊂ X be thelargest open subset such that (U,∆|U) is a simple normal crossing pair.

Then there is a morphism f : X → X given by a composite of blow-upssuch that

(1) f is an isomorphism over U ,

(2) (X, ∆) is a simple normal crossing pair, where ∆ = f−1∗ ∆ +

Exc(f).

For the precise statements and the proof of Theorem 5.2.16 andTheorem 5.2.17, see [BM] and [BVP]. We also recommend the readerto see [BVP, Section 4, Algorithm for the main theorem].

Finally, we recall Grothendieck’s Quot scheme for the reader’s con-venience. For the details, see, for example, [Ni, Theorem 5.14] and[AltKle, Section 2]. We will use it in the proof of Theorem 5.5.1.

Theorem 5.2.18 (Grothendieck). Let S be a noetherian scheme,let π : X → S be a projective morphism, and let L be a relatively veryample line bundle on X. Then for any coherent OX-module E andany polynomial Φ ∈ Q[λ], the functor QuotΦ,LE/X/S is representable by a

projective S-scheme QuotΦ,LE/X/S.

5.3. Du Bois complexes and Du Bois pairs

In this section, we quickly review Du Bois complexes and Du Boissingularities. For the details, see, for example, [Du], [St], [GNPP,Expose V], [Sa], [PS], [Kv5], and [Ko13, Chapter 6].

5.3.1 (Du Bois complexes). Let X be an algebraic variety. Then wecan associate a filtered complex (Ω•X , F ) called the Du Bois complex ofX in a suitable derived category Db

diff,coh(X) (see [Du, 1. Complexes

166 5. INJECTIVITY AND VANISHING THEOREMS

filtres d’operateurs differentiels d’ordre ≤ 1] and Remark 5.3.2 below).We put

Ω0X = Gr0

F Ω•X .

There is a natural map (Ω•X , σ) → (Ω•X , F ). It induces OX → Ω0X .

If OX → Ω0X is a quasi-isomorphism, then X is said to have Du Bois

singularities. We sometimes simply say that X is Du Bois. Let Σbe a reduced closed subvariety of X. Then there is a natural mapρ : (Ω•X , F ) → (Ω•Σ, F ) in Db

diff,coh(X). By taking the cone of ρ with

a shift by one, we obtain a filtered complex (Ω•X,Σ, F ) in Dbdiff,coh(X).

Note that (Ω•X,Σ, F ) was essentially introduced by Steenbrink in [St,Section 3]. We put

Ω0X,Σ = Gr0

F Ω•X,Σ.

Then there are a map JΣ → Ω0X,Σ, where JΣ is the defining ideal sheaf

of Σ on X, and the following commutative diagram

JΣ//

OX //

OΣ+1 //

Ω0X,Σ

// Ω0X

// Ω0Σ

+1 //

in the derived category Dbcoh(X) (see also Remark 5.3.4 below).

For completeness, we include the definitions of the derived cate-gories Db

coh(X), Dbdiff,coh(X), and so on.

Remark 5.3.2 (Derived categories). Let X be a variety. ThenD(X) denotes the derived category of OX-modules and Db

coh(X) is thefull subcategory of D(X) consisting of complexes whose cohomologiesare all coherent and vanish in sufficiently negative and positive degrees.For the details, see [Har1].

Let us consider the category Cdiff(X). Each object of Cdiff(X) is atriple (K•, d, F ) consisting of a complex (K•, d) of OX-modules and adecreasing filtration F on K• such that

(i) K• is bounded below,(ii) the filtration F is biregular, that is, for each component Ki of

K•, there exist integers m and n such that FmKi = Ki andF nKi = 0,

(iii) d is a differential operator of order at most one and preservesthe filtration F , and

(iv) GrpF (d) : GrpF (Ki) → GrpF (Ki+1) is OX-linear for any integersp and i.

5.3. DU BOIS COMPLEXES AND DU BOIS PAIRS 167

Let Ddiff(X) be the derived category of the category Cdiff(X). For thedetails, see [Du]. In this situation, Db

diff,coh(X) is the full subcategoryof Ddiff(X) consisting of (K•, d, F ) such that GrpF (K•) is an object ofDb

coh(X) for every p.

By using the theory of mixed Hodge structures on cohomology withcompact support, we have the following theorem.

Theorem 5.3.3. Let X be a variety and let Σ be a reduced closedsubvariety of X. We put j : X − Σ → X. Then we have the followingproperties.

(1) The complex (Ω•X,Σ)an is a resolution of j!CXan−Σan.(2) If in addition X is proper, then the spectral sequence

Ep,q1 = Hq(X,Ωp

X,Σ)⇒ Hp+q(Xan, j!CXan−Σan)

degenerates at E1, where ΩpX,Σ = GrpF Ω•X,Σ[p].

From now on, we will simply write X (resp. OX and so on) toexpress Xan (resp. OXan and so on) if there is no risk of confusion.

Proof. Here, we use the formulation of [PS, §3.3 and §3.4]. Weassume that X is proper. We take cubical hyperresolutions πX : X• →X and πΣ : Σ• → Σ fitting in a commutative diagram.

Σ•

πΣ

// X•

πX

Σ ι

// X

Let Hdg(X) := RπX∗Hdg•(X•) be a mixed Hodge complex of sheaveson X giving the natural mixed Hodge structure on H•(X,Z) (see[PS, Definition 5.32 and Theorem 5.33]). We can obtain a mixedHodge complex of sheaves Hdg(Σ) := RπΣ∗Hdg•(Σ•) on Σ analo-gously. Roughly speaking, by forgetting the weight filtration and theQ-structure of Hdg(X) and considering it in Db

diff,coh(X), we obtainthe Du Bois complex (Ω•X , F ) of X (see [GNPP, Expose V (3.3)Theoreme]). We can also obtain the Du Bois complex (Ω•Σ, F ) of Σanalogously. By taking the mixed cone of Hdg(X) → ι∗Hdg(Σ) witha shift by one, we obtain a mixed Hodge complex of sheaves on Xgiving the natural mixed Hodge structure on H•c (X − Σ,Z) (see [PS,5.5 Relative Cohomology]). Roughly speaking, by forgetting the weightfiltration and the Q-structure, we obtain the desired filtered complex(Ω•X,Σ, F ) in Db

diff,coh(X). When X is not proper, we take completions

of X and Σ of X and Σ and apply the above arguments to X and Σ.Then we restrict everything to X. The properties (1) and (2) obviously

168 5. INJECTIVITY AND VANISHING THEOREMS

hold by the above description of (Ω•X,Σ, F ). By the above construction

and description of (Ω•X,Σ, F ), we know that the map JΣ → Ω0X,Σ in

Dbcoh(X) is induced by natural maps of complexes. Remark 5.3.4. Note that the Du Bois complex Ω•X is nothing but

the filtered complex RπX∗(Ω•X• , F ). For the details, see [GNPP, Ex-

pose V (3.3) Theoreme and (3.5) Definition]. Therefore, the Du Boiscomplex of the pair (X,Σ) is given by

Cone•(RπX∗(Ω•X• , F )→ ι∗RπΣ∗(Ω

•Σ• , F ))[−1].

By the construction of Ω•X , there is a natural map aX : OX → Ω•Xwhich induces OX → Ω0

X in Dbcoh(X). Moreover, the composition of

aanX : OXan → (Ω•X)an with the natural inclusion CXan ⊂ OXan in-

duces a quasi-isomorphism CXan'−→ (Ω•X)an. We have a natural map

aΣ : OΣ → Ω•Σ with the same properties as aX and the following com-mutative diagram.

OX //

aX

OΣ

aΣ

Ω•X // Ω•Σ

Therefore, we have a natural map b : JΣ → Ω•X,Σ such that b induces

JΣ → Ω0X,Σ in Db

coh(X) and that the composition of ban : (JΣ)an →(Ω•X,Σ)an with the natural inclusion j!CXan−Σan ⊂ (JΣ)an induces a

quasi-isomorphism j!CXan−Σan'−→ (Ω•X,Σ)an. We need the weight fil-

tration and the Q-structure in order to prove the E1-degeneration ofHodge to de Rham type spectral sequence. We used the frameworkof [PS, §3.3 and §3.4] because we had to check that various diagramsrelated to comparison morphisms are commutative (see [PS, Remark3.23]) for the proof of Theorem 5.3.3 (2) and so on.

Let us recall the definition of Du Bois pairs by [Kv5, Definition3.13].

Definition 5.3.5 (Du Bois pairs). With the notation of 5.3.1 andTheorem 5.3.3, if the map JΣ → Ω0

X,Σ is a quasi-isomorphism, thenthe pair (X,Σ) is called a Du Bois pair.

By the definitions, we can easily check the following useful propo-sition.

Proposition 5.3.6. With the notation of 5.3.1 and Theorem 5.3.3,we assume that both X and Σ are Du Bois. Then the pair (X,Σ) is aDu Bois pair, that is, JΣ → Ω0

X,Σ is a quasi-isomorphism.

5.4. HODGE THEORETIC INJECTIVITY THEOREMS 169

Let us recall the following well-known results on Du Bois singular-ities.

Theorem 5.3.7. Let X be a normal algebraic variety with onlyquotient singularities. Then X is Du Bois. Note that X has onlyrational singularities.

Theorem 5.3.7 follows from, for example, [Du, 5.2. Theoreme],[Kv1], and so on.

Lemma 5.3.8. Let X be a variety with closed subvarieties X1 andX2 such that X = X1 ∪ X2. Assume that X1, X2, and X1 ∩ X2 areDu Bois. Note that, in particular, we assume that X1 ∩X2 is reduced.Then X is Du Bois.

For the proof of Lemma 5.3.8, see, for example, [Schw, Lemma3.4].

Although it is dispensable, we will use the notion of Du Bois com-plexes for the proof of the Hodge theoretic injectivity theorem: Theo-rem 5.4.1.

5.4. Hodge theoretic injectivity theorems

The main theorem of this section is:

Theorem 5.4.1 (Hodge theoretic injectivity theorem, see [F36,Theorem 1.1]). Let (X,∆) be a simple normal crossing pair such thatX is proper and that ∆ is a boundary R-divisor on X. Let L be aCartier divisor on X and let D be an effective Weil divisor on X whosesupport is contained in Supp ∆. Assume that L ∼R KX + ∆. Then thenatural homomorphism

Hq(X,OX(L))→ Hq(X,OX(L+D))

induced by the inclusion OX → OX(D) is injective for every q.

Theorem 5.4.1 is nothing but [F36, Theorem 1.1]. As a very usefulspecial case of Theorem 5.4.1, we have:

Theorem 5.4.2 (see [Am2, Theorem 2.3] and [F36, Theorem 1.4]).Let X be a proper smooth algebraic variety and let ∆ be a boundary R-divisor on X such that Supp ∆ is a simple normal crossing divisoron X. Let L be a Cartier divisor on X and let D be an effectiveCartier divisor on X whose support is contained in Supp ∆. Assumethat L ∼R KX + ∆. Then the natural homomorphism

Hq(X,OX(L))→ Hq(X,OX(L+D))

induced by the inclusion OX → OX(D) is injective for every q.

170 5. INJECTIVITY AND VANISHING THEOREMS

Theorem 5.4.2 is a very useful generalization of Lemma 3.1.1. It issufficient for many applications of the minimal model program. How-ever, we need Theorem 5.4.1 for the theory of quasi-log schemes dis-cussed in Chapter 6. For various applications of Theorem 5.4.2, see[F36, Section 5].

First, let us prove Theorem 5.4.2.

Proof of Theorem 5.4.2. Without loss of generality, we mayassume that X is connected. We set S = b∆c and B = ∆. Byperturbing B, we may assume that B is a Q-divisor (cf. Lemma 5.2.13).We setM = OX(L−KX − S). Let N be the smallest positive integersuch that NL ∼ N(KX +S+B). In particular, NB is an integral Weildivisor. We take the N -fold cyclic cover

π′ : Y ′ = SpecX

N−1⊕i=0

M−i → X

associated to the sectionNB ∈ |MN |. More precisely, let s ∈ H0(X,MN)be a section whose zero divisor is NB. Then the dual of s : OX →MN

defines an OX-algebra structure on⊕N−1

i=0 M−i. Let Y → Y ′ be thenormalization and let π : Y → X be the composition morphism. It iswell known that

Y = SpecX

N−1⊕i=0

M−i(biBc).

For the details, see [EsVi3, 3.5. Cyclic covers]. Note that Y hasonly quotient singularities by construction. We set T = π∗S. LetT =

∑i∈I Ti be the irreducible decomposition. Then every irreducible

component of Ti1 ∩ · · · ∩ Tik has only quotient singularities for everyi1, · · · , ik ⊂ I. Hence it is easy to see that both Y and T have onlyDu Bois singularities by Theorem 5.3.7 and Lemma 5.3.8 (see also [I]).Therefore, the pair (Y, T ) is a Du Bois pair by Proposition 5.3.6. Thismeans that OY (−T ) → Ω0

Y,T is a quasi-isomorphism (see also [FFS,3.4]). We note that T is Cartier. Hence OY (−T ) is the defining idealsheaf of T on Y . The E1-degeneration of

Ep,q1 = Hq(Y,Ωp

Y,T )⇒ Hp+q(Y, j!CY−T )

implies that the homomorphism

Hq(Y, j!CY−T )→ Hq(Y,OY (−T ))

induced by the natural inclusion

j!CY−T ⊂ OY (−T )

5.4. HODGE THEORETIC INJECTIVITY THEOREMS 171

is surjective for every q (see Remark 5.3.4). By taking a suitable directsummand

C ⊂M−1(−S)

of

π∗(j!CY−T ) ⊂ π∗OY (−T ),

we obtain a surjection

Hq(X, C)→ Hq(X,M−1(−S))

induced by the natural inclusion C ⊂ M−1(−S) for every q. We cancheck the following simple property by examining the monodromy ac-tion of the Galois group Z/NZ of π : Y → X on C around SuppB.

Lemma 5.4.3 (cf. [KoMo, Corollary 2.54]). Let U ⊂ X be a con-nected open set such that U ∩ Supp ∆ 6= ∅. Then H0(U, C|U) = 0.

Proof of Lemma 5.4.3. If U ∩ SuppB 6= ∅, then H0(U, C|U) =0 since the monodromy action on C around SuppB is nontrivial. IfU ∩ SuppS 6= ∅, then H0(U, C|U) = 0 since C is a direct summand ofπ∗(j!CY−T ) and T = π∗S.

This property is utilized by the following fact. The proof of Lemma5.4.4 is obvious.

Lemma 5.4.4 (cf. [KoMo, Lemma 2.55]). Let F be a sheaf ofAbelian groups on a topological space V and let F1 and F2 be subsheavesof F . Let Z be a closed subset of V . Assume that

(1) F2|V−Z = F |V−Z, and(2) if U is connected, open and U ∩Z 6= ∅, then H0(U, F1|U) = 0.

Then F1 is a subsheaf of F2.

As a corollary, we obtain:

Corollary 5.4.5 (cf. [KoMo, Corollary 2.56]). Let M ⊂M−1(−S)be a subsheaf such that M |X−Supp∆ =M−1(−S)|X−Supp∆. Then the in-jection

C →M−1(−S)

factors as

C →M →M−1(−S).

Therefore,

Hq(X,M)→ Hq(X,M−1(−S))

is surjective for every q.

172 5. INJECTIVITY AND VANISHING THEOREMS

Proof of Corollary 5.4.5. The first part is clear from Lemma5.4.3 and Lemma 5.4.4. This implies that we have maps

Hq(X, C)→ Hq(X,M)→ Hq(X,M−1(−S)).

As we saw above, the composition is surjective. Hence so is the mapon the right.

Therefore, Hq(X,M−1(−S−D))→ Hq(X,M−1(−S)) is surjectivefor every q. By Serre duality, we obtain that

Hq(X,OX(KX)⊗M(S))→ Hq(X,OX(KX)⊗M(S +D))

is injective for every q. This means that

Hq(X,OX(L))→ Hq(X,OX(L+D))

is injective for every q.

Next, let us prove Theorem 5.4.1, the main theorem of this section.The proof of Theorem 5.4.2 given above works for Theorem 5.4.1 withsome minor modifications.

Proof of Theorem 5.4.1. Without loss of generality, we mayassume that X is connected. We can take an effective Cartier divisorD′ onX such thatD′−D is effective and SuppD′ ⊂ Supp ∆. Therefore,by replacing D with D′, we may assume that D is a Cartier divisor.We set S = b∆c and B = ∆. By Lemma 5.2.13, we may assumethat B is a Q-divisor. We set M = OX(L −KX − S). Let N be thesmallest positive integer such that NL ∼ N(KX + S + B). We define

an OX-algebra structure of⊕N−1

i=0 M−i(biBc) by s ∈ H0(X,MN) with(s = 0) = NB. We set

π : Y = SpecX

N−1⊕i=0

M−i(biBc)→ X

and T = π∗S. Let Y =∑

j∈J Yj be the irreducible decomposition.Then every irreducible component of Yj1∩· · ·∩Yjl has only quotient sin-gularities for every j1, · · · , jl ⊂ J by construction. Let T =

∑i∈I Ti

be the irreducible decomposition. Then every irreducible component ofTi1 ∩ · · · ∩ Tik has only quotient singularities for every i1, · · · , ik ⊂ Iby construction. Hence it is easy to see that both Y and T are DuBois by Theorem 5.3.7 and Lemma 5.3.8 (see also [I]). Therefore, thepair (Y, T ) is a Du Bois pair by Proposition 5.3.6. This means thatOY (−T ) → Ω0

Y,T is a quasi-isomorphism (see also [FFS, 3.4]). We

5.4. HODGE THEORETIC INJECTIVITY THEOREMS 173

note that T is Cartier. Hence OY (−T ) is the defining ideal sheaf of Ton Y . The E1-degeneration of

Ep,q1 = Hq(Y,Ωp

Y,T )⇒ Hp+q(Y, j!CY−T )

implies that the homomorphism

Hq(Y, j!CY−T )→ Hq(Y,OY (−T ))

induced by the natural inclusion

j!CY−T ⊂ OY (−T )

is surjective for every q (see Remark 5.3.4). By taking a suitable directsummand

C ⊂M−1(−S)

of

π∗(j!CY−T ) ⊂ π∗OY (−T ),

we obtain a surjection

Hq(X, C)→ Hq(X,M−1(−S))

induced by the natural inclusion C ⊂ M−1(−S) for every q. It is easyto see that Lemma 5.4.3 holds for this new setting. Hence Corollary5.4.5 also holds without any modifications. Therefore,

Hq(X,M−1(−S −D))→ Hq(X,M−1(−S))

is surjective for every q. By Serre duality, we obtain that

Hq(X,OX(L))→ Hq(X,OX(L+D))

is injective for every q. We close this section with an easy application of Theorem 5.4.1.

Corollary 5.4.6 (Kodaira vanishing theorem for simple normalcrossing varieties). Let X be a projective simple normal crossing varietyand let L be an ample line bundle on X. Then Hq(X,OX(KX)⊗L) = 0for every q > 0.

Proof. We take a general member ∆ ∈ |Ll| for some positivelarge number l. Then we can find a Cartier divisor M on X such thatM ∼Q KX + 1

l∆ and that OX(KX)⊗L ' OX(M). Then, by Theorem

5.4.1,

Hq(X,OX(M))→ Hq(X,OX(M +m∆))

is injective for every q and any positive integer m. Since ∆ is ample,Serre’s vanishing theorem implies that Hq(X,OX(M)) = 0 for everyq > 0.

174 5. INJECTIVITY AND VANISHING THEOREMS

5.5. Relative Hodge theoretic injectivity theorem

In this section, we generalize Theorem 5.4.1 for the relative setting.It is much more useful than Theorem 5.4.1.

Theorem 5.5.1 (Relative Hodge theoretic injectivity theorem, see[F36, Theorem 6.1]). Let (X,∆) be a simple normal crossing pair suchthat ∆ is a boundary R-divisor on X. Let π : X → V be a propermorphism between schemes and let L be a Cartier divisor on X andlet D be an effective Weil divisor on X whose support is contained inSupp ∆. Assume that L ∼R,π KX + ∆, that is, there is an R-Cartierdivisor B on V such that L ∼R KX + ∆ + π∗B. Then the naturalhomomorphism

Rqπ∗OX(L)→ Rqπ∗OX(L+D)

induced by the inclusion OX → OX(D) is injective for every q.

By using [BVP] (see Theorem 5.2.17 and Lemma 5.5.2), we canreduce Theorem 5.5.1 to Theorem 5.4.1.

Lemma 5.5.2. Let f : Z → W be a proper morphism from a sim-ple normal crossing pair (Z,∆) to a scheme W . Let W be a projec-tive scheme such that W contains W as a Zariski open dense sub-set. Then there exist a proper simple normal crossing pair (Z,∆) thatis a compactification of (Z,∆) and a proper morphism f : Z → Wthat compactifies f : Z → W . Moreover, Z \ Z is a divisor on Z,Supp ∆ ∪ Supp(Z \ Z) is a simple normal crossing divisor on Z, andZ \Z has no common irreducible components with ∆. We note that wecan make ∆ a K-Cartier K-divisor on Z when so is ∆ on Z, where Kis Z, Q, or R. When f is projective, we can make Z projective.

Proof of Lemma 5.5.2. Let ∆ ⊂ Z be any compactification of∆ ⊂ Z. By blowing up Z inside Z \ Z, we may assume that f :Z → W extends to f : Z → W , Z is a simple normal crossing variety,and Z \ Z is of pure codimension one (see Theorem 5.2.16 and [BM,Theorem 1.5]). By Theorem 5.2.17 (see also [BVP, Theorem 1.4]), wecan construct a desired compactification. Note that we can make ∆ aK-Cartier K-divisor by the argument in [BVP, Section 8].

Remark 5.5.3. We put X = (x2 − zy2 = 0) ⊂ C3. Then X \ 0is a normal crossing variety (see Definition 5.8.3 below). In this case,there is no normal crossing complete variety which contains X \ 0 asa Zariski open subset. For the details, see [F12, 3.6 Whitney umbrella].Therefore, we can not directly apply the arguments in this section to

5.5. RELATIVE HODGE THEORETIC INJECTIVITY THEOREM 175

normal crossing varieties. For the details of the injectivity, torsion-free, and vanishing theorems for normal crossing pairs, see Section 5.8below.

Let us start the proof of Theorem 5.5.1.

Proof of Theorem 5.5.1. By shrinking V , we may assume thatV is affine and L ∼R KX + ∆. Without loss of generality, we mayassume that X is connected. Let V be a projective compactificationof V . By Lemma 5.5.2, we can compactify π : X → V to π : X → V .We put b∆c = S and B = ∆. Let B (resp. S) be the closure ofB (resp. S) on X. We may assume that S is Cartier and B is R-Cartier (see Lemma 5.5.2). We construct a coherent sheaf F on Xwhich is an extension of OX(L). We consider Grothendieck’s Quot

scheme Quot1,OX

F/X/X (see Theorem 5.2.18). Note that the restriction of

Quot1,OX

F/X/X to X is nothing but X because F|X = OX(L) is a line

bundle on X. Therefore, the structure morphism from Quot1,OX

F/X/X to

X has a section s over X. By taking the closure of s(X) in Quot1,OX

F/X/X ,

we have a compactification X† of X and a line bundle L on X† withL|X = OX(L). If necessary, we take more blow-ups of X† outside Xby Theorem 5.2.17 (see also [BVP, Theorem 1.4]). Then we obtain anew compactification X and a Cartier divisor L on X with L|X = L(cf. Lemma 5.5.2). In this situation, L ∼R (KX+∆), where ∆ = S+B,does not necessarily hold. We can write∑

i

bi(fi) = L− (KY + ∆),

where bi is a real number and fi ∈ Γ(X,K∗X) for every i. We set

E =∑i

bi(fi)− (L− (KX + ∆)).

We note that we can see fi ∈ Γ(X,K∗X

) for every i (cf. [Li, Section 7.1Proposition 1.15]). Then we have

L+ dEe ∼R KX + ∆ + −E.

By the above construction, it is obvious that SuppE ⊂ X \X. Let Dbe the closure of D in X. It is sufficient to prove that the map

ϕq : Rqπ∗OX(L+ dEe)→ Rqπ∗OX(L+ dEe+D)

induced by the natural inclusion OX → OX(D) is injective for everyq. Suppose that ϕq is not injective for some q. Let A be a sufficiently

176 5. INJECTIVITY AND VANISHING THEOREMS

ample general Cartier divisor on V such that H0(V ,Kerϕq⊗OV (A)) 6=0. In this case, the map

H0(V ,Rqπ∗OX(L+ dEe)⊗OV (A))

→ H0(V ,Rqπ∗OX(L+ dEe+D)⊗OV (A))

induced by ϕq is not injective. Since A is sufficiently ample, this impliesthat

Hq(X,OX(L+ dEe+ π∗A))

→ Hq(X,OX(L+ dEe+ π∗A+D))

is not injective. Since

L+ dEe+ π∗A ∼R KX + ∆ + −E+ π∗A,

this contradicts Theorem 5.4.1. Hence ϕq is injective for every q.

5.6. Injectivity, vanishing, and torsion-free theorems

The next lemma is an easy generalization of the vanishing theoremof Reid–Fukuda type for simple normal crossing pairs, which is a veryspecial case of Theorem 5.6.3 (i). However, we need Lemma 5.6.1 forour proof of Theorem 5.6.3.

Lemma 5.6.1 (Relative vanishing lemma). Let f : Y → X be aproper morphism from a simple normal crossing pair (Y,∆) to a schemeX such that ∆ is a boundary R-divisor on Y . We assume that f isan isomorphism at the generic point of any stratum of the pair (Y,∆).Let L be a Cartier divisor on Y such that L ∼R,f KY + ∆. ThenRqf∗OY (L) = 0 for every q > 0.

Proof. By shrinking X, we may assume that L ∼R KY + ∆. Byapplying Lemma 5.2.13 to ∆, we may further assume that ∆ is aQ-divisor and L ∼Q KY + ∆.

Step 1. We assume that Y is irreducible. In this case, L−(KY +∆)is nef and log big over X with respect to the pair (Y,∆), that is,L− (KY + ∆) is nef and big over X and (L− (KY + ∆))|W is big overf(W ) for every log canonical center W of the pair (Y,∆) (see Definition3.2.10 and Definition 5.7.2 below). Therefore, Rqf∗OY (L) = 0 for everyq > 0 by the vanishing theorem of Reid–Fukuda type (see, for example,Theorem 3.2.11).

Step 2. Let Y1 be an irreducible component of Y and let Y2 be theunion of the other irreducible components of Y . Then we have a shortexact sequence

0→ OY1(−Y2|Y1)→ OY → OY2 → 0.

5.6. INJECTIVITY, VANISHING, AND TORSION-FREE THEOREMS 177

We set L′ = L|Y1 − Y2|Y1 . Then we have a short exact sequence

0→ OY1(L′)→ OY (L)→ OY2(L|Y2)→ 0

and L′ ∼Q KY1 + ∆|Y1 . On the other hand, we can check that

L|Y2 ∼Q KY2 + Y1|Y2 + ∆|Y2 .

We have already known that Rqf∗OY1(L′) = 0 for every q > 0 by Step

1. By induction on the number of the irreducible components of Y , wehave Rqf∗OY2(L|Y2) = 0 for every q > 0. Therefore, Rqf∗OY (L) = 0for every q > 0 by the exact sequence:

· · · → Rqf∗OY1(L′)→ Rqf∗OY (L)→ Rqf∗OY2(L|Y2)→ · · · .

So, we finish the proof of Lemma 5.6.1. It is the time to state the main injectivity theorem for simple normal

crossing pairs. Our formulation of Theorem 5.6.2 is indispensable forthe proof of our main theorem: Theorem 5.6.3.

Theorem 5.6.2 (Injectivity theorem for simple normal crossingpairs). Let (X,∆) be a simple normal crossing pair such that ∆ is aboundary R-divisor on X and let π : X → V be a proper morphismbetween schemes. Let L be a Cartier divisor on X and let D be an effec-tive Cartier divisor that is permissible with respect to (X,∆). Assumethe following conditions.

(i) L ∼R,π KX + ∆ +H,(ii) H is a π-semi-ample R-divisor, and(iii) tH ∼R,π D+D′ for some positive real number t, where D′ is an

effective R-Cartier R-divisor that is permissible with respect to(X,∆).

Then the homomorphisms

Rqπ∗OX(L)→ Rqπ∗OX(L+D),

which are induced by the natural inclusion OX → OX(D), are injectivefor all q.

Theorem 5.6.2 is new and is a relative version of [F32, Theorem3.4].

Proof of Theorem 5.6.2. We set S = b∆c andB = ∆ through-out this proof. We obtain a projective birational morphism f : Y → Xfrom a simple normal crossing variety Y such that f is an isomorphismover X \ Supp(D + D′ + B), and that the union of the support off ∗(S+B+D+D′) and the exceptional locus of f has a simple normalcrossing support on Y by Theorem 5.2.17 (see also [BVP, Theorem

178 5. INJECTIVITY AND VANISHING THEOREMS

1.4]). Let B′ be the strict transform of B on Y . We may assumethat SuppB′ is disjoint from any strata of Y that are not irreduciblecomponents of Y by taking blow-ups. We write

KY + S ′ +B′ = f ∗(KX + S +B) + E,

where S ′ is the strict transform of S and E is f -exceptional. By theconstruction of f : Y → X, S ′ is Cartier andB′ is R-Cartier. Therefore,E is also R-Cartier. It is easy to see that E+ = dEe ≥ 0. We setL′ = f ∗L + E+ and E− = E+ − E ≥ 0. We note that E+ is Cartierand E− is R-Cartier because SuppE is simple normal crossing on Y(cf. Remark 5.2.12). Without loss of generality, we may assume that Vis affine. Since f∗H is an R>0-linear combination of semi-ample Cartierdivisors, we can write f ∗H ∼R

∑i aiHi, where 0 < ai < 1 and Hi is a

general Cartier divisor on Y for every i. We set

B′′ = B′ + E− +ε

tf∗(D +D′) + (1− ε)

∑i

aiHi

for some 0 < ε 1. Then L′ ∼R KY + S ′ + B′′. By construction,bB′′c = 0, the support of S ′ +B′′ is simple normal crossing on Y , andSuppB′′ ⊃ Supp f ∗D. So, Theorem 5.5.1 implies that the homomor-phisms

Rq(π f)∗OY (L′)→ Rq(π f)∗OY (L′ + f ∗D)

are injective for all q. By Lemma 5.6.1, Rqf∗OY (L′) = 0 for every q > 0and it is easy to see that f∗OY (L′) ' OX(L). By the Leray spectralsequence, the homomorphisms

Rqπ∗OX(L)→ Rqπ∗OX(L+D)

are injective for all q.

Since we formulated Theorem 5.6.2 in the relative setting, the proofof Theorem 5.6.3, which is nothing but [F32, Theorem 1.1], is muchsimpler than the proof given in [F32].

Theorem 5.6.3 (Vanishing and torsion-free theorem for simple nor-mal crossing pairs, see [F32, Theorem 1.1]). Let (Y,∆) be a simplenormal crossing pair such that ∆ is a boundary R-divisor on Y . Letf : Y → X be a proper morphism to a scheme X and let L be a Cartierdivisor on Y such that L− (KY + ∆) is f -semi-ample. Let q be an ar-bitrary non-negative integer. Then we have the following properties.

(i) Every associated prime of Rqf∗OY (L) is the generic point ofthe f -image of some stratum of (Y,∆).

5.6. INJECTIVITY, VANISHING, AND TORSION-FREE THEOREMS 179

(ii) Let π : X → V be a projective morphism to a scheme V suchthat

L− (KY + ∆) ∼R f∗H

for some π-ample R-divisor H on X. Then Rqf∗OY (L) isπ∗-acyclic, that is,

Rpπ∗Rqf∗OY (L) = 0

for every p > 0.

Proof of Theorem 5.6.3 (i). Without loss of generality, we mayassume that X is affine. Suppose that Rqf∗OY (L) has a local sec-tion whose support does not contain the f -images of any strata of(Y,∆). More precisely, let U be a non-empty Zariski open set and lets ∈ Γ(U,Rqf∗OY (L)) be a non-zero section of Rqf∗OY (L) on U whosesupport V ⊂ U does not contain the f -images of any strata of (Y,∆).Without loss of generality, we may further assume that U is affine andX = U by shrinking X. Then we can find a Cartier divisor A on Xwith the following properties:

(a) f∗A is permissible with respect to (Y,∆), and(b) Rqf∗OY (L)→ Rqf∗OY (L)⊗OX(A) is not injective.

This contradicts Theorem 5.6.2. Therefore, the support of every non-zero local section of Rqf∗OY (L) contains the f -image of some stratumof (Y,∆), equivalently, the support of every non-zero local section ofRqf∗OY (L) is equal to the union of the f -images of some strata of(Y,∆). This means that every associated prime of Rqf∗OY (L) is thegeneric point of the f -image of some stratum of (Y,∆).

From now on, we prove Theorem 5.6.3 (ii).

Proof of Theorem 5.6.3 (ii). Without loss of generality, we mayassume that V is affine. In this case, we can write H ∼R H1+H2, whereH1 (resp. H2) is a π-ample Q-divisor (resp. a π-ample R-divisor) on X.So, we can write H2 ∼R

∑i aiAi, where 0 < ai < 1 and Ai is a gen-

eral very ample Cartier divisor over V on X for every i. Replacing B(resp. H) with B +

∑i aif

∗Ai (resp. H1), we may assume that H is aπ-ample Q-divisor. We take a general member A ∈ |mH|, where m isa sufficiently large and divisible positive integer, such that A′ = f ∗Aand Rqf∗OY (L + A′) is π∗-acyclic for all q. By Theorem 5.6.3 (i), wehave the following short exact sequences

0→ Rqf∗OY (L)→ Rqf∗OY (L+ A′)→ Rqf∗OA′(L+ A′)→ 0.

for all q. Note that Rqf∗OA′(L+A′) is π∗-acyclic by induction on dimXand that Rqf∗OY (L + A′) is also π∗-acyclic by the above assumption.

180 5. INJECTIVITY AND VANISHING THEOREMS

Thus, Ep,q2 = 0 for p ≥ 2 in the following commutative diagram of

spectral sequences.

Ep,q2 = Rpπ∗R

qf∗OY (L)

ϕpq

+3 Rp+q(π f)∗OY (L)

ϕp+q

Ep,q

2 = Rpπ∗Rqf∗OY (L+ A′) +3 Rp+q(π f)∗OY (L+ A′)

We note that ϕ1+q is injective by Theorem 5.6.2. We have that

E1,q2

α−→ R1+q(π f)∗OY (L)

is injective by the fact that Ep,q2 = 0 for p ≥ 2. We also have that

E1,q

2 = 0 by the above assumption. Therefore, we obtain E1,q2 = 0 since

the injection

E1,q2

α−→ R1+q(π f)∗OY (L)ϕ1+q

−→ R1+q(π f)∗OY (L+ A′)

factors through E1,q

2 = 0. This implies that Rpπ∗Rqf∗OY (L) = 0 for

every p > 0. As an application of Theorem 5.6.3, we have:

Theorem 5.6.4 (Kodaira vanishing theorem for log canonical pairs,see [F18, Theorem 4.4]). Let (X,∆) be a log canonical pair such that∆ is a boundary R-divisor on X. Let L be a Q-Cartier Weil divisor onX such that L− (KX +∆) is π-ample, where π : X → V is a projectivemorphism. Then Rqπ∗OX(L) = 0 for every q > 0.

Proof. Let f : Y → X be a resolution of singularities of X suchthat

KY = f ∗(KX + ∆) +∑i

aiEi

with ai ≥ −1 for every i. We may assume that∑

iEi ∪ Supp f∗L is asimple normal crossing divisor on Y . We put

E =∑i

aiEi

andF =

∑aj=−1

(1− bj)Ej,

where bj = multEjf ∗L. We note that A = L− (KX + ∆) is π-ample

by assumption. We have

f ∗A = f ∗L− f∗(KX + ∆)

= df ∗L+ E + F e − (KY + F + −(f∗L+ E + F )).

5.6. INJECTIVITY, VANISHING, AND TORSION-FREE THEOREMS 181

We can easily check that

f∗OY (df ∗L+ E + F e) ' OX(L)

and that F + −(f ∗L+E +F ) has a simple normal crossing supportand is a boundary R-divisor on Y . By Theorem 5.6.3 (ii), we obtainthat OX(L) is π∗-acyclic. Thus, we have Rqπ∗OX(L) = 0 for everyq > 0.

We note that Theorem 5.6.4 contains a complete form of [Kv2,Theorem 0.3] as a corollary. For the related topics, see [KSS, Corollary1.3].

Corollary 5.6.5 (Kodaira vanishing theorem for log canonicalvarieties). Let X be a projective log canonical variety and let L be anample Cartier divisor on X. Then

Hq(X,OX(KX + L)) = 0

for every q > 0. Furthermore, if we assume that X is Cohen–Macaulay,then Hq(X,OX(−L)) = 0 for every q < dimX.

Remark 5.6.6. We can see that Corollary 5.6.5 is contained in [F6,Theorem 2.6], which is a very special case of Theorem 5.6.3 (ii). Theauthor forgot to state Corollary 5.6.5 explicitly in [F6]. There, we donot need embedded simple normal crossing pairs.

Note that there are typos in the proof of [F6, Theorem 2.6]. In thecommutative diagram, Rif∗ωX(D)’s should be replaced byRjf∗ωX(D)’s.

We close this section with an easy example.

Example 5.6.7. Let X be a projective log canonical threefoldwhich has the following properties: (i) there exists a projective bira-tional morphism f : Y → X from a smooth projective threefold, and(ii) the exceptional locus E of f is an Abelian surface with KY =f ∗KX−E. For example, X is a cone over a normally projective Abeliansurface in PN and f : Y → X is the blow-up at the vertex of X. Let Lbe an ample Cartier divisor on X. By the Leray spectral sequence, wehave

0→ H1(X, f∗f∗OX(−L))→ H1(Y, f ∗OX(−L))

→ H0(X,R1f∗f∗OX(−L))→ H2(X, f∗f

∗OX(−L))

→ H2(Y, f ∗OX(−L))→ · · · .

Therefore, we obtain

H2(X,OX(−L)) ' H0(X,OX(−L)⊗R1f∗OY ),

182 5. INJECTIVITY AND VANISHING THEOREMS

becauseH1(Y, f ∗OX(−L)) = H2(Y, f ∗OX(−L)) = 0 by the Kawamata–Viehweg vanishing theorem (see Theorem 3.2.1). On the other hand,we have

Rqf∗OY ' Hq(E,OE)

for every q > 0 since Rqf∗OY (−E) = 0 for every q > 0 by the Grauert–Riemenschneider vanishing theorem (see Theorem 3.2.7). Thus, weobtain H2(X,OX(−L)) ' C2. In particular, H2(X,OX(−L)) 6= 0.We note that X is not Cohen–Macaulay. In the above example, if weassume that E is a K3-surface, then Hq(X,OX(−L)) = 0 for q < 3and X is Cohen–Macaulay. For the details, see Section 7.2, especially,Lemma 7.2.7.

5.7. Vanishing theorems of Reid–Fukuda type

Here, we treat some generalizations of Theorem 5.6.3. First, weintroduce the notion of nef and log big divisors.

Definition 5.7.1 (Nef and log big divisors). Let f : (Y,∆) → Xbe a proper morphism from a simple normal crossing pair (Y,∆) to ascheme X. Let π : X → V be a proper morphism between schemesand let H be an R-Cartier divisor on X. We say that H is nef and logbig over V with respect to f : (Y,∆)→ X if and only if H|C is nef andbig over V for any C, where C is the image of a stratum of (Y,∆).

We also need the notion of nef and log big divisors for normal pairs.

Definition 5.7.2 (Nef and log big divisors for normal pairs). Let(X,∆) be a normal pair and let π : X → V be a proper morphism.Let D be an R-Cartier divisor on X. We say that D is nef and log bigover V with respect to (X,∆) if and only if D is nef and big over Vand D|C is big over V for every log canonical center C of (X,∆).

We can generalize Theorem 5.6.3 as follows. It is [Am1, Theorem7.4] for embedded simple normal crossing pairs.

Theorem 5.7.3 (cf. [Am1, Theorem 7.4]). Let f : (Y,∆) → Xbe a proper morphism from an embedded simple normal crossing pair(Y,∆) to a scheme X such that ∆ is a boundary R-divisor. Let L be aCartier divisor on Y and let π : X → V be a proper morphism betweenschemes. Assume that

f∗H ∼R L− (KY + ∆),

where H is nef and log big over V with respect to f : (Y,∆) → X.Let q be an arbitrary non-negative integer. Then we have the followingproperties.

5.7. VANISHING THEOREMS OF REID–FUKUDA TYPE 183

(i) Every associated prime of Rqf∗OY (L) is the generic point ofthe f -image of some stratum of (Y,∆).

(ii) We have

Rpπ∗Rqf∗OY (L) = 0

for every p > 0.

Proof. Note that L− (KY +∆) is f -semi-ample. Therefore, (i) isa special case of Theorem 5.6.3 (i).

From now on, we will prove (ii). We note that we may assume thatV is affine without loss of generality.

Step 1. We assume that every stratum of (Y,∆) dominates someirreducible component of X. By taking the Stein factorization, we mayassume that f has connected fibers. Then we may further assume thatX is irreducible and every stratum of (Y,∆) dominates X. By Chow’slemma, there exists a projective birational morphism µ : X ′ → Xsuch that π′ : X ′ → V is projective. By taking a proper birationalmorphism ϕ : Y ′ → Y that is an isomorphism over the generic pointof any stratum of (Y,∆), we have the following commutative diagram.

Y ′ϕ //

g

Y

f

X ′

π′ BBB

BBBB

B µ// X

π

V

Then, by Theorem 5.2.17 (see also [BVP, Theorem 1.4]), we can write

KY ′ + ∆′ = ϕ∗(KY + ∆) + E,

where

(1) (Y ′,∆′) is an embedded simple normal crossing pair such that∆′ is a boundary R-divisor.

(2) E is an effective ϕ-exceptional Cartier divisor.(3) Every stratum of (Y ′,∆′) dominates X ′.

We note that every stratum of (Y,∆) dominates X. Therefore,

ϕ∗L+ E ∼R KY ′ + ∆′ + ϕ∗f ∗H.

We note that

ϕ∗OY ′(ϕ∗L+ E) ' OY (L)

and

Riϕ∗OY ′(ϕ∗L+ E) = 0

184 5. INJECTIVITY AND VANISHING THEOREMS

for every i > 0 by Theorem 5.6.3 (i). Thus, by replacing Y and L withY ′ and ϕ∗L+E, we may assume that ϕ : Y ′ → Y is the identity, thatis, we have

Y

g

Y

f

X ′

µ //

π′ BBB

BBBB

B X

π

V.

We put F = Rqg∗OY (L). Since µ∗H is nef and big over V and π′ :X ′ → V is projective, we can write µ∗H = E+A, where A is a π′-ampleR-divisor on X ′ and E is an effective R-Cartier R-divisor by Kodaira(see Lemma 2.1.18). By the same arguments as above, we take someblow-ups and may further assume that (Y,∆ + g∗E) is an embeddedsimple normal crossing pair. If k is a sufficiently large positive integer,then

b∆+1

kg∗Ec = 0,

µ∗H =1

kE +

1

kA+

k − 1

kµ∗H,

and1

kA+

k − 1

kµ∗H

is π′-ample. Thus, F is µ∗-acyclic and (πµ)∗ = π′∗-acyclic by Theorem5.6.3 (ii). We note that

L−(KY + ∆ +

1

kg∗E

)∼R g

∗(1

kA+

k − 1

kµ∗H

).

So, we have Rqf∗OY (L) ' µ∗F and Rqf∗OY (L) is π∗-acyclic. Thus,we finish the proof when every stratum of (Y,∆) dominates some irre-ducible component of X.

Step 2. We treat the general case by induction on dim f(Y ). Bytaking some embedded log transformations (see Lemma 5.7.4 below),we can decompose Y = Y ′ ∪ Y ′′ as follows: Y ′ is the union of all strataof (Y,∆) that are not mapped to irreducible components of X andY ′′ = Y − Y ′. We put

KY ′′ + ∆Y ′′ = (KY + ∆)|Y ′′ − Y ′|Y ′′ .

Then f : (Y ′′,∆Y ′′)→ X and L′′ = L|Y ′′−Y ′|Y ′′ satisfy the assumptionin Step 1. We consider the following short exact sequence

0→ OY ′′(L′′)→ OY (L)→ OY ′(L)→ 0.

5.7. VANISHING THEOREMS OF REID–FUKUDA TYPE 185

By taking Rqf∗, we have short exact sequence

0→ Rqf∗OY ′′(L′′)→ Rqf∗OY (L)→ Rqf∗OY ′(L)→ 0

for every q. This is because the connecting homomorphisms

Rqf∗OY ′(L)→ Rq+1f∗OY ′′(L′′)

are zero maps for every q by (i). Since (ii) holds for the first and thirdmembers by Step 1 and by induction on the dimension, respectively, italso holds for Rqf∗OY (L).

So, we finish the proof. We have already used Lemma 5.7.4 in the proof of Theorem 5.7.3.

Lemma 5.7.4 is easy to check. So we omit the proof.

Lemma 5.7.4 (cf. [Am1, p.218 embedded log transformation]). Let(X,∆) be an embedded simple normal crossing pair and let M be anambient space of (X,∆). Let C be a smooth stratum of (X,∆). Letσ : N →M be the blow-up along C. Let Y denote the reduced structureof the total transform of X in N . We put

KY + ∆Y = f ∗(KX + ∆),

where f = σ|Y . Then we have the following properties.

(i) (Y,∆Y ) is an embedded simple normal crossing pair with anambient space N .

(ii) f∗OY ' OX and Rif∗OY = 0 for every i > 0.(iii) The strata of (X,∆) are exactly the images of the strata of

(Y,∆Y ).(iv) σ−1(C) is a maximal (with respect to the inclusion) stratum of

(Y,∆Y ).(v) If ∆ is a boundary R-divisor on X, then ∆Y is a boundary

R-divisor on Y .

Remark 5.7.5. We need the notion of embedded simple normalcrossing pairs to prove Theorem 5.7.3 even when Y is smooth. It isa key point of the proof of Theorem 5.7.3. Note that we do not needthe assumption that Y is embedded in Step 1 in the proof of Theorem5.7.3.

As a corollary of Theorem 5.7.3, we can prove the following van-ishing theorem. It is the culmination of the works of several au-thors: Kawamata, Viehweg, Nadel, Reid, Fukuda, Ambro, Fujino, andothers. To the author’s best knowledge, we can not find it in theliterature except [F17]. Note that Theorem 5.7.6 is a complete gener-alization of [KMM, Theorem 1-2-5].

186 5. INJECTIVITY AND VANISHING THEOREMS

Theorem 5.7.6 (see [F17, Theorem 2.48]). Let (X,∆) be a logcanonical pair such that ∆ is a boundary R-divisor and let L be a Q-Cartier Weil divisor on X. Assume that L − (KX + ∆) is nef andlog big over V with respect to (X,∆), where π : X → V is a propermorphism. Then Rqπ∗OX(L) = 0 for every q > 0.

Proof. Let f : Y → X be a log resolution of (X,∆) such that

KY = f ∗(KX + ∆) +∑i

aiEi

with ai ≥ −1 for every i. We may assume that∑

iEi ∪ Supp f∗L is asimple normal crossing divisor on Y . We put

E =∑i

aiEi

and

F =∑aj=−1

(1− bj)Ej,

where bj = multEjf ∗L. We note that A = L− (KX + ∆) is nef and

log big over V with respect (X,∆) by assumption. So, we have

f ∗A = f ∗L− f∗(KX + ∆)

= df ∗L+ E + F e − (KY + F + −(f∗L+ E + F )).

We can easily check that

f∗OY (df ∗L+ E + F e) ' OX(L)

and that F + −(f ∗L+E +F ) has a simple normal crossing supportand is a boundary R-divisor on Y . By the above definition of F , A is nefand log big over V with respect to f : (Y, F +−(f∗L+E+F ))→ X.Therefore, by Theorem 5.7.3 (ii), we obtain that OX(L) is π∗-acyclic.Thus, we have Rqπ∗OX(L) = 0 for every q > 0.

As a special case, we have the Kawamata–Viehweg vanishing theo-rem for klt pairs.

Corollary 5.7.7 (Kawamata–Viehweg vanishing theorem, see [KMM,Remark 1-2-6]). Let (X,∆) be a klt pair and let L be a Q-Cartier Weildivisor on X. Assume that L− (KX +∆) is nef and big over V , whereπ : X → V is a proper morphism. Then Rqπ∗OX(L) = 0 for everyq > 0.

We add one example.

5.7. VANISHING THEOREMS OF REID–FUKUDA TYPE 187

Example 5.7.8. Let Y be a projective surface which has the fol-lowing properties: (i) there exists a projective birational morphismf : X → Y from a smooth projective surface X, and (ii) the ex-ceptional locus E of f is an elliptic curve with KX + E = f∗KY . Forexample, Y is a cone over a smooth plane cubic curve and f : X → Yis the blow-up at the vertex of Y . We note that (X,E) is a plt pair.Let H be an ample Cartier divisor on Y . We consider a Cartier divisorL = f ∗H +KX +E on X. Then L− (KX +E) is nef and big, but notlog big with respect to (X,E). By the short exact sequence

0→ OX(f∗H +KX)→ OX(f∗H +KX + E)→ OE(KE)→ 0,

we obtain

R1f∗OX(f ∗H +KX + E) ' H1(E,OE(KE)) ' C(P ),

where P = f(E). By the Leray spectral sequence, we have

0→ H1(Y, f∗OX(KX + E)⊗OY (H))→ H1(X,OX(L))

→ H0(Y,C(P ))→ H2(Y, f∗OX(KX + E)⊗OY (H))

→ · · · .

If H is sufficiently ample, then H1(X,OX(L)) ' H0(Y,C(P )) ' C(P ).In particular, H1(X,OX(L)) 6= 0.

Remark 5.7.9. In Example 5.7.8, there exists an effective Q-divisorB on X such that L− 1

kB is ample for every k > 0 by Kodaira’s lemma

(see Lemma 2.1.18). Since L ·E = 0, we have B ·E < 0. In particular,

(X,E +1

kB)

is not log canonical for any k > 0. This is the main reason whyH1(X,OX(L)) 6= 0. If (X,E + 1

kB) were log canonical, then the am-

pleness of L − (KX + E + 1kB) would imply H1(X,OX(L)) = 0 by

Theorem 5.6.4.

If Y is quasi-projective in Theorem 5.7.3, we do not need the as-sumption that the pair (Y,∆) is embedded.

Theorem 5.7.10 ([FF, Theorem 6.3]). Let f : (Y,∆) → X be aproper morphism from a quasi-projective simple normal crossing pair(Y,∆) to a scheme X such that ∆ is a boundary R-divisor. Let L be aCartier divisor on Y and let π : X → V be a proper morphism betweenschemes. Assume that

f∗H ∼R L− (KY + ∆),

188 5. INJECTIVITY AND VANISHING THEOREMS

where H is nef and log big over V with respect to f : (Y,∆) → X.Let q be an arbitrary non-negative integer. Then we have the followingproperties.

(ii) We haveRpπ∗R

qf∗OY (L) = 0

for every p > 0.

We can easily reduce Theorem 5.7.10 to Theorem 5.7.3. For theproof of Theorem 5.7.3, see the proof of [FF, Theorem 6.3]. We usedTheorem 5.7.10 for the proof of the main theorem of [FF].

5.8. From SNC pairs to NC pairs

In this section, we prove the injectivity, vanishing, and torsion-free theorems for embedded normal crossing pairs, although the resultsin this section are not necessary for the theory of quasi-log schemesdiscussed in Chapter 6.

Theorem 5.8.1 is a generalization of [Am1, Theorem 3.1].

Theorem 5.8.1. Let (X,∆) be an embedded normal crossing pairsuch that ∆ is a boundary R-divisor and let π : X → V be a propermorphism between schemes. Let L be a Cartier divisor on X and let Dbe an effective Cartier divisor that is permissible with respect to (X,∆).Assume the following conditions.

(i) L ∼R,π KX + ∆ +H,(ii) H is a π-semi-ample R-divisor, and(iii) tH ∼R,π D+D′ for some positive real number t, where D′ is an

effective R-Cartier R-divisor that is permissible with respect to(X,∆).

Then the homomorphisms

Rqπ∗OX(L)→ Rqπ∗OX(L+D),

which are induced by the natural inclusion OX → OX(D), are injectivefor all q.

Theorem 5.8.2 is nothing but [Am1, Theorem 7.4].

Theorem 5.8.2. Let (Y,∆) be an embedded normal crossing pairsuch that ∆ is a boundary R-divisor. Let f : Y → X be a propermorphism between schemes and let L be a Cartier divisor on Y suchthat L− (KY +∆) is f -semi-ample. Let q be an arbitrary non-negativeinteger. Then we have the following properties.

5.8. FROM SNC PAIRS TO NC PAIRS 189

(ii) Let π : X → V be a proper morphism between schemes. Weassume that

L− (KY + ∆) ∼R f∗H

where H is an R-Cartier divisor on X which is nef and log bigover V with respect to f : (Y,∆) → X. Then we obtain thatRqf∗OY (L) is π∗-acyclic, that is,

Rpπ∗Rqf∗OY (L) = 0

for every p > 0.

Before we go to the proof, let us recall the definition of normalcrossing pairs. The following definition is the same as [Am1, Definition2.3] though it may look different.

Definition 5.8.3 (Normal crossing pair). A variety X has normalcrossing singularities if, for every closed point x ∈ X,

OX,x 'C[[x0, · · · , xN ]]

(x0 · · ·xk)for some 0 ≤ k ≤ N , where N = dimX. Let X be a normal crossingvariety. We say that a reduced divisor D on X is normal crossing if,in the above notation, we have

OD,x 'C[[x0, · · · , xN ]]

(x0 · · ·xk, xi1 · · ·xil)for some i1, · · · , il ⊂ k+ 1, · · · , N. We say that the pair (X,∆) isa normal crossing pair if the following conditions are satisfied.

(1) X is a normal crossing variety, and(2) ∆ is an R-Cartier R-divisor whose support is normal crossing

on X.

We say that a normal crossing pair (X,∆) is embedded if there existsa closed embedding ι : X → M , where M is a smooth variety ofdimension dimX + 1. We call M the ambient space of (X,∆). We put

KXν + Θ = ν∗(KX + ∆),

where ν : Xν → X is the normalization of X. A stratum of (X,∆) isan irreducible component of X or the ν-image of some log canonicalcenter of (Xν ,Θ) on X.

A Cartier divisor B on a normal crossing pair (X,∆) is called per-missible with respect to (X,∆) if the support of B contains no strata of

190 5. INJECTIVITY AND VANISHING THEOREMS

the pair (X,∆). A finite Q-linear (resp. R-linear) combination of per-missible Cartier divisor with respect to (X,∆) is called a permissibleQ-divisor (resp. R-divisor) with respect to (X,∆).

The following definition is almost obvious.

Definition 5.8.4 (Nef and log big divisors). Let f : (Y,∆) → Xbe a proper morphism from a normal crossing pair (Y,∆) to a schemeX. Let π : X → V be a proper morphism between schemes and let Hbe an R-Cartier divisor on X. We say that H is nef and log big overV with respect to f : (Y,∆)→ X if and only if H|C is nef and big overV for any C, where C is the image of a stratum of (Y,∆).

The following three lemmas are easy to check. So, we omit theproofs.

Lemma 5.8.5. Let X be a normal crossing divisor on a smoothvariety M . Then there exists a sequence of blow-ups

Mk →Mk−1 → · · · →M0 = M

with the following properties.

(i) σi+1 : Mi+1 →Mi is the blow-up along a smooth stratum of Xi

for every i ≥ 0,(ii) X0 = X and Xi+1 is the inverse image of Xi with the reduced

structure for every i ≥ 0, and(iii) Xk is a simple normal crossing divisor on Mk.

For each step σi+1, we can directly check that

σi+1∗OXi+1' OXi

and

Rqσi+1∗OXi+1= 0

for every i ≥ 0 and q ≥ 1. Let ∆ be an R-Cartier R-divisor on X suchthat Supp ∆ is normal crossing. We put ∆0 = ∆ and

KXi+1+ ∆i+1 = σ∗i+1(KXi

+ ∆i)

for all i ≥ 0. Then it is obvious that ∆i is an R-Cartier R-divisor andSupp ∆i is normal crossing on Xi for every i ≥ 0. We can also checkthat ∆i is a boundary R-divisor (resp. a boundary Q-divisor) for everyi ≥ 0 if so is ∆. If ∆ is a boundary R-divisor, then the σi+1-image ofany stratum of (Xi+1,∆i+1) is a stratum of (Xi,∆i).

Remark 5.8.6. Each step in Lemma 5.8.5 is called embedded logtransformation in [Am1, Section 2]. See also Lemma 5.7.4.

5.8. FROM SNC PAIRS TO NC PAIRS 191

Lemma 5.8.7. Let X be a simple normal crossing divisor on asmooth variety M . Let S+B be a boundary R-Cartier R-divisor on Xsuch that Supp(S +B) is normal crossing, S is reduced, and bBc = 0.Then there exists a sequence of blow-ups

Mk →Mk−1 → · · · →M0 = M

with the following properties.

(i) σi+1 : Mi+1 → Mi is the blow-up along a smooth stratum of(Xi, Si) that is contained in Si for every i ≥ 0,

(ii) we put X0 = X, S0 = S, and B0 = B, and Xi+1 is the stricttransform of Xi for every i ≥ 0,

(iii) we define

KXi+1+ Si+1 +Bi+1 = σ∗i+1(KXi

+ Si +Bi)

for every i ≥ 0, where Bi+1 is the strict transform of Bi onXi+1,

(iv) the σi+1-image of any stratum of (Xi+1, Si+1 +Bi+1) is a stra-tum of (Xi, Si +Bi), and

(v) Sk is a simple normal crossing divisor on Xk.

For each step σi+1, we can easily check that

σi+1∗OXi+1' OXi

and

Rqσi+1∗OXi+1= 0

for every i ≥ 0 and q ≥ 1. We note that Xi is simple normal crossing,Supp(Si + Bi) is normal crossing on Xi, and Si is reduced for everyi ≥ 0.

Lemma 5.8.8. Let X be a simple normal crossing divisor on asmooth variety M . Let S + B be a boundary R-Cartier R-divisor onX such that Supp(S + B) is normal crossing, S is reduced and sim-ple normal crossing, and bBc = 0. Then there exists a sequence ofblow-ups

Mk →Mk−1 → · · · →M0 = M

with the following properties.

(i) σi+1 : Mi+1 → Mi is the blow-up along a smooth stratum of(Xi, SuppBi) that is contained in SuppBi for every i ≥ 0,

(ii) we put X0 = X, S0 = S, and B0 = B, and Xi+1 is the stricttransform of Xi for every i ≥ 0,

(iii) we define

KXi+1+ Si+1 +Bi+1 = σ∗i+1(KXi

+ Si +Bi)

192 5. INJECTIVITY AND VANISHING THEOREMS

for every i ≥ 0, where Si+1 is the strict transform of Si onXi+1, and

(iv) Supp(Sk +Bk) is a simple normal crossing divisor on Xk.

We note that Xi is simple normal crossing on Mi and Supp(Si + Bi)is normal crossing on Xi for every i ≥ 0. We can easily check thatbBic ≤ 0 for every i ≥ 0. The composition morphism Mk → M isdenoted by σ. Let L be any Cartier divisor on X. We put E = d−Bke.Then E is an effective σ-exceptional Cartier divisor on Xk and weobtain

σ∗OXk(σ∗L+ E) ' OX(L)

and

Rqσ∗OXk(σ∗L+ E) = 0

for every q ≥ 1 by Theorem 5.6.3 (i). We note that

σ∗L+ E − (KXk+ Sk + Bk) = σ∗L− σ∗(KX + S +B)

is R-linearly trivial over X and σ is an isomorphism at the genericpoint of any stratum of (Xk, Sk +Bk).

Let us go to the proof of Theorems 5.8.1 and 5.8.2.

Proof of Theorem 5.8.1. We take a sequence of blow-ups andobtain a projective morphism σ : X ′ → X from an embedded simplenormal crossing variety X ′ by Lemma 5.8.5. We can replace X andL with X ′ and σ∗L by Leray’s spectral sequence. So, we may assumethat X is simple normal crossing. We put S = b∆c and B = ∆.Similarly, we may assume that S is simple normal crossing on X byapplying Lemma 5.8.7. Finally, we use Lemma 5.8.8 and obtain abirational morphism

σ : (X ′, S ′ +B′)→ (X,S +B)

from an embedded simple normal crossing pair (X ′, S ′ +B′) such that

KX′ + S ′ +B′ = σ∗(KX + S +B)

as in Lemma 5.8.8. By Lemma 5.8.8, we can replace (X,S + B) andL with (X ′, S ′ + B′) and σ∗L+ d−B′e by Leray’s spectral sequence.Then we apply Theorem 5.6.2. Thus, we obtain Theorem 5.8.1.

Proof of Theorem 5.8.2. We take a sequence of blow-ups andobtain a projective morphism σ : Y ′ → Y from an embedded simplenormal crossing variety Y ′ by Lemma 5.8.5. We can replace Y andL with Y ′ and σ∗L by Leray’s spectral sequence. So, we may assumethat Y is simple normal crossing. We put S = b∆c and B = ∆.Similarly, we may assume that S is simple normal crossing on Y by

5.9. EXAMPLES 193

applying Lemma 5.8.7. Finally, we use Lemma 5.8.8 and obtain abirational morphism

σ : (Y ′, S ′ +B′)→ (Y, S +B)

from an embedded simple normal crossing pair (Y ′, S ′ +B′) such that

KY ′ + S ′ +B′ = σ∗(KY + S +B)

as in Lemma 5.8.8. By Lemma 5.8.8, we can replace (Y, S + B) andL with (Y ′, S ′ + B′) and σ∗L+ d−B′e by Leray’s spectral sequence.Then we apply Theorem 5.7.3. Thus, we obtain Theorem 5.8.2.

5.9. Examples

In this section, we treat various supplementary examples. Theseexamples show that some results obtained in this chapter are sharp.

5.9.1 (Hodge theoretic injectivity theorems). Let X be a smoothprojective variety and let M be a Cartier divisor on X such that N ∼mM , where N is a reduced simple normal crossing divisor on X andm ≥ 2. We put ∆ = 1

mN and L = KX + M . In this setting, we can

apply Theorem 5.4.2 since L ∼Q KX +∆. If M is semi-ample, then theexistence of such N and m is obvious by Bertini. Here, we give someexplicit examples where M is not nef.

Example 5.9.2. We consider the P1-bundle

π : X = PP1(OP1 ⊕OP1(2))→ P1.

Let E and G be the sections of π such that E2 = −2 and G2 = 2. Wenote that E+2F ∼ G, where F is a fiber of π. We consider M = E+F .Then

2M = 2E + 2F ∼ E +G.

In this case, M · E = −1. In particular, M is not nef. Unfortunately,we can easily check that

H i(X,OX(KX +M)) = 0

for every i. So, it is not interesting to apply Theorem 5.4.2.

Example 5.9.3. We consider the P1-bundle

π : Y = PP1(OP1 ⊕OP1(4))→ P1.

Let G (resp. E) be the positive (resp. negative) section of π, that is,the section corresponding to the projection OP1 ⊕ OP1(4) → OP1(4)(resp. OP1 ⊕ OP1(4) → OP1). We put M ′ = −F + 2G, where F is afiber of π. Then M ′ is not nef and

2M ′ ∼ G+ E + F1 + F2 +H,

194 5. INJECTIVITY AND VANISHING THEOREMS

where F1 and F2 are distinct fibers of π, and H is a general memberof the free linear system |2G|. Note that G + E + F1 + F2 + H is areduced simple normal crossing divisor on Y . We put X = Y × C,where C is an elliptic curve, and M = p∗M ′, where p : X → Y is theprojection. Then X is a smooth projective variety and M is a Cartierdivisor on X. We note that M is not nef and that we can find a reducedsimple normal crossing divisor N such that N ∼ 2M . By the Kunnethformula, we have

H1(X,OX(KX +M)) ' H0(P1,OP1(1)) ' C2.

Therefore, X with L = KX +M and ∆ = 12N satisfies the conditions

in Theorem 5.4.2. Moreover, we have H1(X,OX(L)) 6= 0.

Example 5.9.2 shows that the assumptions for the Hodge theoreticinjectivity theorems in Section 5.4 are geometric.

5.9.4 (Kodaira vanishing theorem for singular varieties). The fol-lowing example is due to Sommese (cf. [Som, (0.2.4) Example]). Itshows that the Kodaira vanishing theorem does not necessarily holdfor varieties with non-lc singularities.

Proposition 5.9.5 (Sommese). We consider the P3-bundle

π : Y = PP1(OP1 ⊕OP1(1)⊕3)→ P1

over P1. LetM = OY (1) be the tautological line bundle of π : Y → P1.We take a general member X of the linear system |(M⊗π∗OP1(−1))⊗4|.Then X is a normal projective Gorenstein threefold and X is not logcanonical. We put L =M⊗ π∗OP1(1). Then L is ample. In this case,we can check that H2(X,L−1) = C. By Serre duality,

H1(X,OX(KX)⊗ L) = C.Therefore, the Kodaira vanishing theorem does not hold for X.

Proof. We consider the following short exact sequence

0→ L−1(−X)→ L−1 → L−1|X → 0.

Then we have the long exact sequence

· · · → H i(Y,L−1(−X))→ H i(Y,L−1)→ H i(X,L−1)

→ H i+1(Y,L−1(−X))→ · · · .Since H i(Y,L−1) = 0 for i < 4 by the original Kodaira vanishingtheorem (see Theorem 3.1.3), we obtain that

H2(X,L−1) = H3(Y,L−1(−X)).

Therefore, it is sufficient to prove that H3(Y,L−1(−X)) = C.

5.9. EXAMPLES 195

We have

L−1(−X) =M−1 ⊗ π∗OP1(−1)⊗M−4 ⊗ π∗OP1(4)

=M−5 ⊗ π∗OP1(3).

We note that Riπ∗M−5 = 0 for i 6= 3 because M = OY (1). ByGrothendieck duality,

RHom(Rπ∗M−5,OP1(KP1)[1]) = Rπ∗RHom(M−5,OY (KY )[4]).

By Grothendieck duality again,

Rπ∗M−5 = RHom(Rπ∗RHom(M−5,OY (KY )[4]),OP1(KP1)[1])

= RHom(Rπ∗(OY (KY )⊗M5),OP1(KP1))[−3]

= (∗).By definition, we have

OY (KY ) = π∗(OP1(KP1)⊗ det(OP1 ⊕OP1(1)⊕3))⊗M−4

= π∗OP1(1)⊗M−4.

By this formula, we obtain

OY (KY )⊗M5 = π∗OP1(1)⊗M.

Thus, Riπ∗(OY (KY )⊗M5) = 0 for every i > 0. We note that

π∗(OY (KY )⊗M5) = OP1(1)⊗ π∗M= OP1(1)⊗ (OP1 ⊕OP1(1)⊕3)

= OP1(1)⊕OP1(2)⊕3.

Therefore, we have

(∗) = RHom(OP1(1)⊕OP1(2)⊕3,OP1(−2))[−3]

= (OP1(−3)⊕OP1(−4)⊕3)[−3].

So, we obtain R3π∗M−5 = OP1(−3)⊕OP1(−4)⊕3. Thus, we have

R3π∗M−5 ⊗OP1(3) = OP1 ⊕OP1(−1)⊕3.

By the spectral sequence, we have

H3(Y,L−1(−X)) = H3(Y,M−5 ⊗ π∗OP1(3))

= H0(P1, R3π∗(M−5 ⊗ π∗OP1(3)))

= H0(P1,OP1 ⊕OP1(−1)⊕3)

= C.Therefore, H2(X,L−1) = C.

Let us recall that X is a general member of the linear system

|(M⊗ π∗OP1(−1))⊗4|.

196 5. INJECTIVITY AND VANISHING THEOREMS

Let C be the negative section of π : Y → P1, that is, the sectioncorresponding to the projection

OP1 ⊕OP1(1)⊕3 → OP1 → 0.

From now, we will check that |M⊗π∗OP1(−1)| is free outside C. Oncewe checked it, we know that |(M⊗ π∗OP1(−1))⊗4| is free outside C.Then X is smooth in codimension one. Since Y is smooth, X is normaland Gorenstein by adjunction.

We take Z ∈ |M⊗ π∗OP1(−1)| 6= ∅. Since

H0(Y,M⊗ π∗OP1(−1)⊗ π∗OP1(−1)) = 0,

Z can not have a fiber of π as an irreducible component, that is, anyirreducible component of Z is mapped onto P1 by π : Y → P1. On theother hand, let l be a line in a fiber of π : Y → P1. Then Z · l = 1.Therefore, Z is irreducible. Let F = P3 be a fiber of π : Y → P1. Weconsider

0 = H0(Y,M⊗ π∗OP1(−1)⊗OY (−F ))→ H0(Y,M⊗ π∗OP1(−1))

→ H0(F,OF (1))→ H1(Y,M⊗ π∗OP1(−1)⊗OY (−F ))→ · · · .Since (M⊗ π∗OP1(−1)) · C = −1, every member of |M⊗ π∗OP1(−1)|contains C. We put P = F ∩ C. Then the image of

α : H0(Y,M⊗ π∗OP1(−1))→ H0(F,OF (1))

is H0(F,mP ⊗ OF (1)), where mP is the maximal ideal of P . This isbecause the dimension of H0(Y,M⊗ π∗OP1(−1)) is three. Thus, weknow that |M ⊗ π∗OP1(−1)| is free outside C. In particular, |(M⊗π∗OP1(−1))⊗4| is free outside C.

More explicitly, the image of the injection

α : H0(Y,M⊗ π∗OP1(−1))→ H0(F,OF (1))

is H0(F,mP ⊗OF (1)). We note that

H0(Y,M⊗ π∗OP1(−1)) = H0(P1,OP1(−1)⊕O⊕3P1 ) = C3,

and

H0(Y, (M⊗ π∗OP1(−1))⊗4) = H0(P1, Sym4(OP1(−1)⊕O⊕3P1 )) = C15.

We can check that the restriction of H0(Y, (M⊗ π∗OP1(−1))⊗4) to Fis Sym4H0(F,mP ⊗OF (1)). Thus, the general fiber f of π : X → P1 isa cone in P3 on a smooth plane curve of degree 4 with the vertex P =f ∩ C. Therefore, (Y,X) is not log canonical because the multiplicityof X along C is four. Thus, X is not log canonical by the inversion ofadjunction. Anyway, X is the required variety.

By the same construction, we have:

5.9. EXAMPLES 197

Example 5.9.6. We consider the Pk+1-bundle

π : Y = PP1(OP1 ⊕OP1(1)⊕(k+1))→ P1

over P1 for k ≥ 2. We put M = OY (1) and L = M ⊗ π∗OP1(1).Then L is ample. We take a general member X of the linear system|(M⊗π∗OP1(−1))⊗(k+2)|. Then we can check the following properties.

(1) X is a normal projective Gorenstein (k + 1)-fold.(2) X is not log canonical.(3) We can check

Rk+1π∗M−(k+3) = OP1(−1− k)⊕OP1(−2− k)⊕(k+1)

and

Riπ∗M−(k+3) = 0

for i 6= k + 1.(4) Since L−1(−X) =M−(k+3) ⊗ π∗OP1(k + 1), we have

Hk+1(Y,L−1(−X)) = H0(P1, Rk+1π∗M−(k+3) ⊗OP1(k + 1))

= H0(P1,OP1 ⊕OP1(−1)⊕(k+1))

= C.

Thus, Hk(X,L−1) = Hk+1(Y,L−1(−X)) = C.

We note that the first cohomology group of an anti-ample line bun-dle on a normal variety with dimension ≥ 2 always vanishes by thefollowing Mumford vanishing theorem.

Theorem 5.9.7 (Mumford). Let V be a normal complete algebraicvariety and let L be a semi-ample line bundle on V . Assume thatκ(V,L) ≥ 2. Then H1(V,L−1) = 0.

Proof. Let f : W → V be a resolution of singularities. By Leray’sspectral sequence, we obtain

0→ H1(V, f∗f∗L−1)→ H1(W, f ∗L−1)→ · · · .

By the Kawamata–Viehweg vanishing theorem (see Theorem 3.3.7)and Serre duality, H1(W, f ∗L−1) = 0. Thus, we obtain H1(V,L−1) =H1(V, f∗f

∗L−1) = 0.

5.9.8 (On the Kawamata–Viehweg vanishing theorem). The nextexample shows that a naive generalization of the Kawamata–Viehwegvanishing theorem does not necessarily hold for varieties with log canon-ical singularities. Example 5.9.9 is also a supplement to Theorem 5.7.6.

198 5. INJECTIVITY AND VANISHING THEOREMS

Example 5.9.9. We put V = P2 × P2. Let pi : V → P2 be thei-th projection for i = 1 and 2. We define L = p∗1OP2(1) ⊗ p∗2OP2(1)and consider the P1-bundle π : W = PV (L ⊕ OV ) → V . Let F =P2 × P2 be the negative section of π : W → V , that is, the section ofπ corresponding to L ⊕ OV → OV → 0. By using the linear system|OW (1)⊗ π∗p∗1OP2(1)|, we can contract F = P2 × P2 to P2 × point.

Next, we consider an elliptic curve C ⊂ P2 and put Z = C × C ⊂V = P2 × P2. Let π : Y → Z be the restriction of π : W → V to Z.The restriction of the above contraction morphism

Φ|OW (1)⊗π∗p∗1OP2 (1)| : W → U

to Y is denoted by f : Y → X. Then, the exceptional locus of f : Y →X is E = F |Y = C × C and f contracts E to C × point.

LetOW (1) be the tautological line bundle of the P1-bundle π : W →V . By the construction, OW (1) = OW (D), where D is the positivesection of π, that is, the section corresponding to L ⊕ OW → L → 0.By definition,

OW (KW ) = π∗(OV (KV )⊗ L)⊗OW (−2).

By adjunction,

OY (KY ) = π∗(OZ(KZ)⊗ L|Z)⊗OY (−2) = π∗(L|Z)⊗OY (−2).

Therefore,

OY (KY + E) = π∗(L|Z)⊗OY (−2)⊗OY (E).

We note that E = F |Y . Since OY (E) ⊗ π∗(L|Z) ' OY (D), we haveOY (−(KY +E)) = OY (1) because OY (1) = OY (D). Thus, −(KY +E)is nef and big.

On the other hand, it is not difficult to see that X is a normalprojective Gorenstein threefold, X is log canonical but not klt alongG = f(E), and that X is smooth outside G. Since we can check thatf ∗KX = KY + E, −KX is nef and big.

Finally, we consider the short exact sequence

0→ J → OX → OX/J → 0,

where J is the multiplier ideal sheaf of X. In our case, we can easilycheck that J = f∗OY (−E) = IG, where IG is the defining ideal sheafof G on X. Since −KX is nef and big, H i(X,J ) = 0 for every i > 0by Nadel’s vanishing theorem (see Theorem 3.4.2). Therefore,

H i(X,OX) = H i(G,OG)

for every i > 0. Since G is an elliptic curve,

H1(X,OX) = H1(G,OG) = C.

5.9. EXAMPLES 199

We note that −KX is nef and big but −KX is not log big with respectto X.

5.9.10 (On the injectivity theorem). The final example in this sec-tion supplements Theorem 5.6.2.

Example 5.9.11. We consider the P1-bundle

π : X = PP1(OP1 ⊕OP1(1))→ P1.

Let S (resp. H) be the negative (resp. positive) section of π, thatis, the section corresponding to the projection OP1 ⊕ OP1(1) → OP1

(resp. OP1⊕OP1(1)→ OP1(1)). Then H is semi-ample and S+F ∼ H,where F is a fiber of π.

Claim. The homomorphism

H1(X,OX(KX + S +H))→ H1(X,OX(KX + S +H + S + F ))

induced by the natural inclusion OX → OX(S + F ) is not injective.

Proof of Claim. It is sufficient to see that the homomorphism

H1(X,OX(KX + S +H))→ H1(X,OX(KX + S +H + F ))

induced by the natural inclusion OX → OX(F ) is not injective. Weconsider the short exact sequence

0→ OX(KX + S +H)→ OX(KX + S +H + F )

→ OF (KF + (S +H)|F )→ 0.

We note that F ' P1 and OF (KF + (S +H)|F ) ' OP1 . Therefore, weobtain the following exact sequence

0→ C→ H1(X,OX(KX+S+H))→ H1(X,OX(KX+S+H+F ))→ 0.

Thus,

H1(X,OX(KX + S +H))→ H1(X,OX(KX + S +H + F ))

is not injective. We note that S + F is not permissible with respect to(X,S).

Therefore, the permissibility assumption is indispensable for Theo-rem 5.6.2.

CHAPTER 6

Fundamental theorems for quasi-log schemes

This chapter is the main part of this book. In this chapter, we in-troduce the notion of quasi-log schemes and establish the fundamentaltheorems for quasi-log schemes.

Section 6.1 is an overview of the main results of this chapter. InSection 6.2, we introduce the notion of quasi-log schemes. Note that ourtreatment is slightly different from Ambro’s original theory of quasi-log varieties (see [Am1]). In Section 6.3, we discuss various basicproperties, for example, adjunction and vanishing theorems, of quasi-log schemes. In Section 6.4, we show that a normal pair has a naturalgood quasi-log structure. By this fact, we can apply the theory ofquasi-log schemes to normal pairs. We also treat toric polyhedra asexamples of quasi-log schemes. Section 6.5 is devoted to the proof ofthe basepoint-free theorem for quasi-log schemes. In Section 6.6, weprove the rationality theorem for quasi-log schemes. In Section 6.7,we discuss the cone and contraction theorem for quasi-log schemes.Thus we establish the fundamental theorems for quasi-log schemes. InSection 6.8, we discuss some properties of quasi-log Fano schemes andrelated topics. Section 6.9 is devoted to the proof of the basepoint-freetheorem of Reid–Fukuda type for quasi-log schemes. Here, we proveit under some extra assumptions. For the details of the basepoint-freetheorem of Reid–Fukuda type for quasi-log schemes, see the author’srecent preprint [F40].

6.1. Overview

In this chapter, we establish the fundamental theorems for quasi-log schemes. This means that we prove adjunction (see Theorem 6.3.4(i)), various Kodaira type vanishing theorems (see Theorem 6.3.4 (ii)),basepoint-free theorem (see Theorem 6.5.1), rationality theorem (seeTheorem 6.6.1), cone and contraction theorem (see Theorem 6.7.4), andso on, for quasi-log schemes after we introduce the notion of quasi-logschemes. Note that our formulation of the theory of quasi-log schemesis slightly different from Ambro’s original one in [Am1].

In this book, we adopt the following definition of quasi-log schemes.

201

202 6. FUNDAMENTAL THEOREMS FOR QUASI-LOG SCHEMES

Definition 6.1.1 (Quasi-log schemes, see Definition 6.2.2). A quasi-log scheme is a scheme X endowed with an R-Cartier divisor (or R-linebundle) ω on X, a proper closed subscheme X−∞ ⊂ X, and a finitecollection C of reduced and irreducible subschemes of X such thatthere is a proper morphism f : (Y,BY )→ X from a globally embeddedsimple normal crossing pair satisfying the following properties:

(1) f∗ω ∼R KY +BY .(2) The natural map OX → f∗OY (d−(B<1

Y )e) induces an isomor-phism

IX−∞'−→ f∗OY (d−(B<1

Y )e − bB>1Y c),

where IX−∞ is the defining ideal sheaf of X−∞.(3) The collection of subvarieties C coincides with the images

of (Y,BY )-strata that are not included in X−∞.

We simply write [X,ω] to denote the above data(X,ω, f : (Y,BY )→ X

)if there is no risk of confusion. The subvarieties C are called the qlcstrata of [X,ω].

Once we establish the following adjunction and vanishing theo-rem for quasi-log schemes (see Theorem 6.3.4), the notion of quasi-logschemes becomes very useful. Therefore, Theorem 6.1.2 is a key resultof the theory of quasi-log schemes. The proof of Theorem 6.1.2 heavilydepends on the results discussed in Chapter 5.

Theorem 6.1.2 (see Theorem 6.3.4). Let [X,ω] be a quasi-log schemeand let X ′ be the union of X−∞ with a (possibly empty) union of someqlc strata of [X,ω]. Then we have the following properties.

(i) (Adjunction). Assume that X ′ 6= X−∞. Then X ′ is a quasi-logscheme with ω′ = ω|X′ and X ′−∞ = X−∞. Moreover, the qlcstrata of [X ′, ω′] are exactly the qlc strata of [X,ω] that areincluded in X ′.

(ii) (Vanishing theorem). Assume that π : X → S is a propermorphism between schemes. Let L be a Cartier divisor on Xsuch that L−ω is nef and log big over S with respect to [X,ω].Then Riπ∗(IX′⊗OX(L)) = 0 for every i > 0, where IX′ is thedefining ideal sheaf of X ′ on X.

One of the main results of this chapter is:

Theorem 6.1.3 (Cone and contraction theorem, Theorem 6.7.4).Let [X,ω] be a quasi-log scheme and let π : X → S be a projectivemorphism between schemes. Then we have the following properties.

6.2. ON QUASI-LOG SCHEMES 203

(i) We have:

NE(X/S) = NE(X/S)ω≥0 +NE(X/S)−∞ +∑

Rj,

where Rj’s are the ω-negative extremal rays of NE(X/S) thatare rational and relatively ample at infinity. In particular, eachRj is spanned by an integral curve Cj on X such that π(Cj) isa point.

(ii) Let H be a π-ample R-divisor on X. Then there are onlyfinitely many Rj’s included in (ω + H)<0. In particular, theRj’s are discrete in the half-space ω<0.

(iii) Let F be an ω-negative extremal face of NE(X/S) that is rel-atively ample at infinity. Then F is a rational face. In partic-ular, F is contractible at infinity.

We give a proof of Theorem 6.1.3 in Section 6.7 after we establishthe basepoint-free theorem for quasi-log schemes (see Theorem 6.5.1)and the rationality theorem for quasi-log schemes (see Theorem 6.6.1).Note that the proof of the basepoint-free theorem and the rationalitytheorem is based on Theorem 6.1.2.

In Section 6.4, we see that a normal pair has a natural quasi-logstructure. By this fact, we can apply the results in this chapter tonormal pairs.

As we mentioned above, our treatment is slightly different fromAmbro’s original one. So, if the reader wants to taste the originalflavor of the theory of quasi-log varieties, then we recommend him tosee [Am1].

6.2. On quasi-log schemes

First, let us recall the definition of globally embedded simple normalcrossing pairs in order to define quasi-log schemes.

Definition 6.2.1 (Globally embedded simple normal crossing pairs,see [F17, Definition 2.16]). Let Y be a simple normal crossing divisoron a smooth variety M and let D be an R-divisor on M such thatSupp(D+Y ) is a simple normal crossing divisor on M and that D andY have no common irreducible components. We put BY = D|Y andconsider the pair (Y,BY ). We call (Y,BY ) a globally embedded simplenormal crossing pair and M the ambient space of (Y,BY ).

Let us define quasi-log schemes.

Definition 6.2.2 (Quasi-log schemes). A quasi-log scheme is ascheme X endowed with an R-Cartier divisor (or R-line bundle) ω

204 6. FUNDAMENTAL THEOREMS FOR QUASI-LOG SCHEMES

on X, a proper closed subscheme X−∞ ⊂ X, and a finite collectionC of reduced and irreducible subschemes of X such that there is aproper morphism f : (Y,BY ) → X from a globally embedded simplenormal crossing pair satisfying the following properties:

(1) f∗ω ∼R KY +BY .(2) The natural map OX → f∗OY (d−(B<1

Y )e) induces an isomor-phism

IX−∞'−→ f∗OY (d−(B<1

Y )e − bB>1Y c),

where IX−∞ is the defining ideal sheaf of X−∞.(3) The collection of subvarieties C coincides with the images

of (Y,BY )-strata that are not included in X−∞.

We simply write [X,ω] to denote the above data(X,ω, f : (Y,BY )→ X

)if there is no risk of confusion. Note that a quasi-log scheme [X,ω] isthe union of C and X−∞. We also note that ω is called the quasi-logcanonical class of [X,ω], which is defined up to R-linear equivalence. Arelative quasi-log scheme X/S is a quasi-log scheme X endowed with aproper morphism π : X → S. We sometimes simply say that [X,ω] isa quasi-log pair. The subvarieties C are called the qlc strata of [X,ω],X−∞ is called the non-qlc locus of [X,ω], and f : (Y,BY )→ X is calleda quasi-log resolution of [X,ω]. We sometimes use Nqlc(X,ω) to denoteX−∞.

For the details of the various definitions of quasi-log schemes, see[F39, Section 4 and Section 8].

Remark 6.2.3. Let Div(Y ) be the group of Cartier divisors on Yand let Pic(Y ) be the Picard group of Y . Let

δY : Div(Y )⊗ R→ Pic(Y )⊗ R

be the homomorphism induced by A 7→ OY (A) where A is a Cartierdivisor on Y . When ω is an R-line bundle in Definition 6.2.2,

f ∗ω ∼R KY +BY

means

f∗ω = δY (KY +BY )

in Pic(Y )⊗R. Even when ω is an R-line bundle, we usually use −ω todenote the inverse of ω in Pic(X)⊗ R if there is no risk of confusion.

We give an important remark on Definition 6.2.2.

6.2. ON QUASI-LOG SCHEMES 205

Remark 6.2.4 (Schemes versus varieties). A quasi-log scheme inDefinition 6.2.2 is called a quasi-log variety in [Am1] (see also [F17]).However, X is not always reduced when X−∞ 6= ∅ (see Example 6.2.5below). Therefore, we will use the word quasi-log schemes in this paper.Note that X is reduced when X−∞ = ∅ (see Remark 6.2.11 below).

Example 6.2.5 ([Am1, Examples 4.3.4]). Let X be an effectiveCartier divisor on a smooth variety M . Assume that Y , the reducedpart of X, is non-empty. We put ω = (KM + X)|X . Let X−∞ bethe union of the non-reduced components of X. We put KY + BY =(KM +X)|Y . Let f : Y → X be the closed embedding. Then(

X,ω, f : (Y,BY )→ X)

is a quasi-log scheme. Note that X has non-reduced irreducible com-ponents if X−∞ 6= ∅. We also note that f is not surjective if X−∞ 6= ∅.

Remark 6.2.6. A qlc stratum of [X,ω] was originally called a qlccenter of [X,ω] in the literature (see, [Am1], [F17], and so on). Wechanged the terminology.

Definition 6.2.7 (Qlc centers). A closed subvariety C of X iscalled a qlc center of [X,ω] if C is a qlc stratum of [X,ω] which is notan irreducible component of X.

For various applications, the notion of qlc pairs is very useful.

Definition 6.2.8 (Qlc pairs). Let [X,ω] be a quasi-log pair. Wesay that [X,ω] has only quasi-log canonical singularities (qlc singulari-ties, for short) if X−∞ = ∅. Assume that [X,ω] is a quasi-log pair withX−∞ = ∅. Then we sometimes simply say that [X,ω] is a qlc pair.

We give some important remarks on the non-qlc locus X−∞.

Remark 6.2.9. We put A = d−(B<1Y )e and N = bB>1

Y c. Then weobtain the following big commutative diagram.

0 // f∗OY (A−N) // f∗OY (A) // f∗ON(A)

0 // f∗OY (−N) //

α1

OO

f∗OY //

α2

OO

f∗ON

α3

OO

0 // IX−∞//

β1

OO

OX //

β2

OO

OX−∞//

β3

OO

0

Note that αi is a natural injection for every i. By an easy diagramchasing,

IX−∞'−→ f∗OY (A−N)

206 6. FUNDAMENTAL THEOREMS FOR QUASI-LOG SCHEMES

factors through f∗OY (−N). Then we obtain β1 and β3. Since α1 isinjective and α1 β1 is an isomorphism, α1 and β1 are isomorphisms.Therefore, we obtain that f(Y ) ∩ X−∞ = f(N). Note that f is notalways surjective when X−∞ 6= ∅. It sometimes happens that X−∞contains some irreducible components of X. See, for example, Example6.2.5.

Remark 6.2.10 (Semi-normality). By restricting the isomorphism

IX−∞'−→ f∗OY (A−N)

to the open subset U = X \X−∞, we obtain

OU'−→ f∗Of−1(U)(A).

This implies that

OU'−→ f∗Of−1(U)

because A is effective. Therefore, f : f−1(U)→ U is surjective and hasconnected fibers. Note that f−1(U) is a simple normal crossing variety.Thus, U is semi-normal. In particular, U = X \X−∞ is reduced.

Remark 6.2.11. If the pair [X,ω] has only qlc singularities, equiv-alently, X−∞ = ∅, then X is reduced and semi-normal by Remark6.2.10. Note that f(Y ) ∩ X−∞ = ∅ if and only if BY = B≤1

Y , equiva-lently, B>1

Y = 0, by the descriptions in Remark 6.2.9.

We close this section with the definition of nef and log big divisorson quasi-log schemes.

Definition 6.2.12 (Nef and log big divisors on quasi-log schemes).Let L be an R-Cartier divisor (or R-line bundle) on a quasi-log pair[X,ω] and let π : X → S be a proper morphism between schemes.Then L is nef and log big over S with respect to [X,ω] if L is π-nef andL|C is π-big for every qlc stratum C of [X,ω].

6.3. Basic properties of quasi-log schemes

In this section, we discuss some basic properties of quasi-log schemes.Theorem 6.3.4 is the main theorem of this section. Note that Theorem6.3.4 heavily depends on the results discussed in Chapter 5.

The following proposition makes the theory of quasi-log schemesflexible.

Proposition 6.3.1 ([F17, Proposition 3.50]). Let f : V → W be aproper birational morphism between smooth varieties and let BW be an

6.3. BASIC PROPERTIES OF QUASI-LOG SCHEMES 207

R-divisor on W such that SuppBW is a simple normal crossing divisoron W . Assume that

KV +BV = f ∗(KW +BW )

and that SuppBV is a simple normal crossing divisor on V . Then wehave

f∗OV (d−(B<1V )e − bB>1

V c) ' OW (d−(B<1W )e − bB>1

W c).Furthermore, let S be a simple normal crossing divisor on W such thatS ⊂ SuppB=1

W . Let T be the union of the irreducible components ofB=1V that are mapped into S by f . Assume that Supp f−1

∗ BW ∪ Exc(f)is a simple normal crossing divisor on V . Then we have

f∗OT (d−(B<1T )e − bB>1

T c) ' OS(d−(B<1S )e − bB>1

S c),where (KV +BV )|T = KT +BT and (KW +BW )|S = KS +BS.

Proof. By KV +BV = f ∗(KW +BW ), we obtain

KV =f ∗(KW +B=1W + BW)

+ f ∗(bB<1W c+ bB>1

W c)− (bB<1V c+ bB>1

V c)−B=1V − BV .

If a(ν,W,B=1W + BW) = −1 for a prime divisor ν over W , then we

can check that a(ν,W,BW ) = −1 by the same argument as in the proofof Lemma 2.3.9. Since

f∗(bB<1W c+ bB>1

W c)− (bB<1V c+ bB>1

V c)is Cartier, we can easily see that

f∗(bB<1W c+ bB>1

W c) = bB<1V c+ bB>1

V c+ E,

where E is an effective f -exceptional divisor. Thus, we obtain

f∗OV (d−(B<1V )e − bB>1

V c) ' OW (d−(B<1W )e − bB>1

W c).Next, we consider the short exact sequence:

0→ OV (d−(B<1V )e − bB>1

V c − T )

→ OV (d−(B<1V )e − bB>1

V c)→ OT (d−(B<1T )e − bB>1

T c)→ 0.

Since T = f ∗S − F , where F is an effective f -exceptional divisor, wecan easily see that

f∗OV (d−(B<1V )e − bB>1

V c − T ) ' OW (d−(B<1W )e − bB>1

W c − S).

We note that

(d−(B<1V )e − bB>1

V c − T )− (KV + BV +B=1V − T )

= −f∗(KW +BW ).

208 6. FUNDAMENTAL THEOREMS FOR QUASI-LOG SCHEMES

Therefore, every associated prime of R1f∗OV (d−(B<1V )e−bB>1

V c−T ) isthe generic point of the f -image of some stratum of (V, BV +B=1

V −T )(see, for example, Theorem 5.6.3, Theorem 3.16.3, and [F28, Theorem6.3 (i)]).

Claim. No strata of (V, BV +B=1V −T ) are mapped into S by f .

Proof of Claim. Assume that there is a stratum C of (V, BV +B=1V − T ) such that f(C) ⊂ S. Note that

Supp f ∗S ⊂ Supp f−1∗ BW ∪ Exc(f)

andSuppB=1

V ⊂ Supp f−1∗ BW ∪ Exc(f).

Since C is also a stratum of (V,B=1V ) and

C ⊂ Supp f ∗S,

there exists an irreducible component G of B=1V such that

C ⊂ G ⊂ Supp f ∗S.

Therefore, by the definition of T , G is an irreducible component of Tbecause f(G) ⊂ S and G is an irreducible component of B=1

V . So, C isnot a stratum of (V, BV +B=1

V − T ). This is a contradiction. On the other hand, f(T ) ⊂ S. Therefore,

f∗OT (d−(B<1T )e − bB>1

T c)→ R1f∗OV (d−(B<1Z )e − bB>1

Z c − T )

is a zero map by Claim. Thus, we obtain

f∗OT (d−(B<1T )e − bB>1

T c) ' OS(d−(B<1S )e − bB>1

S c)by the following commutative diagram.

0

0

OW (d−(B<1

W )e − bB>1W c − S)

' // f∗OV (d−(B<1V )e − bB>1

V c − T )

OW (d−(B<1

W )e − bB>1W c)

' // f∗OV (d−(B<1V )e − bB>1

V c)

OS(d−(B<1

S )e − bB>1S c)

// f∗OT (d−(B<1T )e − bB>1

T c)

0 0

We finish the proof.

6.3. BASIC PROPERTIES OF QUASI-LOG SCHEMES 209

It is easy to check:

Proposition 6.3.2. In Proposition 6.3.1, let C ′ be a log canonicalcenter of (V,BV ) contained in T . Then f(C ′) is a log canonical centerof (W,BW ) contained in S or f(C ′) is contained in SuppB>1

W . Let C bea log canonical center of (W,BW ) contained in S. Then there exists alog canonical center C ′ of (V,BV ) contained in T such that f(C ′) = C.

The following important theorem is missing in [F17].

Theorem 6.3.3. In Definition 6.2.2, we may assume that the am-bient space M of the globally embedded simple normal crossing pair(Y,BY ) is quasi-projective. In particular, Y is quasi-projective.

Proof. In Definition 6.2.2, we may assume that D + Y is an R-divisor on a smooth variety M such that Supp(D + Y ) is a simplenormal crossing divisor on M , D and Y have no common irreduciblecomponents, and BY = D|Y as in Definition 6.2.1. Let g : M ′ → Mbe a projective birational morphism from a smooth quasi-projectivevariety M ′ with the following properties:

(i) KM ′ +BM ′ = g∗(KM +D + Y ),(ii) SuppBM ′ is a simple normal crossing divisor on M ′, and(iii) Supp g−1

∗ (D + Y ) ∪ Exc(g) is also a simple normal crossingdivisor on M ′.

Let Y ′ be the union of the irreducible components of B=1M ′ that are

mapped into Y by g. We put

(KM ′ +BM ′)|Y ′ = KY ′ +BY ′ .

Then

g∗OY ′(d−(B<1Y ′ )e − bB>1

Y ′ c) ' OY (d−(B<1Y )e − bB>1

Y c)

by Proposition 6.3.1. This implies that

IX−∞'−→ f∗g∗OY ′(d−(B<1

Y ′ )e − bB>1Y ′ c).

By the above construction,

KY ′ +BY ′ = g∗(KY +BY ) ∼R g∗f∗ω.

By Proposition 6.3.2, the collection of subvarieties C in Definition6.2.2 coincides with the images of (Y ′, BY ′)-strata that are not con-tained in X−∞. Therefore, by replacing M and (Y,BY ) with M ′

and (Y ′, BY ′), we may assume that the ambient space M is quasi-projective.

210 6. FUNDAMENTAL THEOREMS FOR QUASI-LOG SCHEMES

The following theorem is a key result for the theory of quasi-logschemes. It follows from the Kollar type torsion-free and vanishingtheorem for simple normal crossing varieties discussed in Chapter 5(see Theorem 5.6.3 and Theorem 5.7.3).

Theorem 6.3.4 (see [Am1, Theorems 4.4 and 7.3] and [F17, The-orem 3.39]). Let [X,ω] be a quasi-log scheme and let X ′ be the union ofX−∞ with a (possibly empty) union of some qlc strata of [X,ω]. Thenwe have the following properties.

(i) (Adjunction). Assume that X ′ 6= X−∞. Then X ′ is a quasi-logscheme with ω′ = ω|X′ and X ′−∞ = X−∞. Moreover, the qlcstrata of [X ′, ω′] are exactly the qlc strata of [X,ω] that areincluded in X ′.

(ii) (Vanishing theorem). Assume that π : X → S is a propermorphism between schemes. Let L be a Cartier divisor on Xsuch that L−ω is nef and log big over S with respect to [X,ω].Then Riπ∗(IX′⊗OX(L)) = 0 for every i > 0, where IX′ is thedefining ideal sheaf of X ′ on X.

Proof. By taking some blow-ups of the ambient spaceM of (Y,BY ),we may assume that the union of all strata of (Y,BY ) mapped to X ′,which is denoted by Y ′, is a union of irreducible components of Y (seeProposition 6.3.1). We putKY ′+BY ′ = (KY +BY )|Y ′ and Y ′′ = Y −Y ′.We prove that f : (Y ′, BY ′)→ X ′ gives the desired quasi-log structureon [X ′, ω′]. By construction, we have f ∗ω′ ∼R KY ′ + BY ′ on Y ′. Weput A = d−(B<1

Y )e and N = bB>1Y c. We consider the following short

exact sequence

0→ OY ′′(−Y ′)→ OY → OY ′ → 0.

By applying ⊗OY (A−N), we have

0→ OY ′′(A−N − Y ′)→ OY (A−N)→ OY ′(A−N)→ 0.

By applying f∗, we obtain

0→ f∗OY ′′(A−N − Y ′)→ f∗OY (A−N)→ f∗OY ′(A−N)

→ R1f∗OY ′′(A−N − Y ′)→ · · · .

By Theorem 5.6.3, no associated prime of R1f∗OY ′′(A − N − Y ′) iscontained in X ′ = f(Y ′). We note that

(A−N − Y ′)|Y ′′ − (KY ′′ + BY ′′+B=1Y ′′ − Y ′|Y ′′) = −(KY ′′ +BY ′′)

∼R −(f∗ω)|Y ′′ ,

6.3. BASIC PROPERTIES OF QUASI-LOG SCHEMES 211

where KY ′′ + BY ′′ = (KY + BY )|Y ′′ . Therefore, the connecting homo-morphism δ : f∗OY ′(A−N)→ R1f∗OY ′′(A−N −Y ′) is zero. Thus weobtain the following short exact sequence

0→ f∗OY ′′(A−N − Y ′)→ IX−∞ → f∗OY ′(A−N)→ 0.

We put IX′ = f∗OY ′′(A−N−Y ′). Then IX′ defines a scheme structureon X ′. We put IX′

−∞= IX−∞/IX′ . Then IX′

−∞' f∗OY ′(A − N) by

the above exact sequence. By the following big commutative diagram:

0 // f∗OY ′′(A−N − Y ′)

// f∗OY (A−N)

// f∗OY ′(A−N) //

0

0 // f∗OY ′′(A− Y ′) // f∗OY (A) // f∗OY ′(A)

0 // IX′

OO

// OX

OO

// OX′ //

OO

0,

we can see that OX′ → f∗OY ′(d−(B<1Y ′ )e) induces an isomorphism

IX′−∞

'−→ f∗OY ′(d−(B<1Y ′ )e − bB>1

Y ′ c).

Therefore, [X ′, ω′] is a quasi-log pair such that X ′−∞ = X−∞. We notethe following big commutative diagram.

0

0

0 // IX′ // IX−∞

// IX′−∞

//

0

0 // IX′ // OX //

OX′

// 0

OX−∞

' // OX′−∞

0 0

By construction, the property on qlc strata is obvious. So, we obtainthe desired quasi-log structure of [X ′, ω′] in (i).

Let f : (Y,BY )→ X be a quasi-log resolution as in the proof of (i).If X ′ = X−∞ in the above proof of (i), then we can easily see that

f∗OY ′′(A−N − Y ′) ' f∗OY ′′(A−N) ' IX−∞ = IX′ .

212 6. FUNDAMENTAL THEOREMS FOR QUASI-LOG SCHEMES

Note that

f ∗(L− ω) ∼R f∗L− (KY ′′ +BY ′′)

on Y ′′, where KY ′′ +BY ′′ = (KY +BY )|Y ′′ . We also note that

f∗L− (KY ′′ +BY ′′)

= (f ∗L+ A−N − Y ′)|Y ′′ − (KY ′′ + BY ′′+B=1Y ′′ − Y ′|Y ′′)

and that no stratum of (Y ′′, BY ′′+B=1Y ′′ −Y ′|Y ′′) is mapped to X−∞.

Then, by Theorem 5.7.3, we have

Riπ∗(f∗OY ′′(f∗L+ A−N − Y ′)) = Riπ∗(IX′ ⊗OX(L)) = 0

for every i > 0. Thus, we obtain the desired vanishing theorem in(ii).

Let us recall the following well-known lemma for the reader’s con-venience (see [Am1, Proposition 4.7] and [F17, Proposition 3.44]).

Lemma 6.3.5. Let [X,ω] be a quasi-log scheme with X−∞ = ∅.Assume that X is the unique qlc stratum of [X,ω]. Then X is normal.

The following proof is different from Ambro’s original one (see[Am1, Proposition 4.7]).

Proof. Let f : (Y,BY ) → X be a quasi-log resolution. SinceX−∞ = ∅, we have f∗OY (d−(B<1

Y )e) ' OX . This implies that f∗OY 'OX . Let ν : Xν → X be the normalization. By assumption, X isirreducible and every stratum of (Y,BY ) is mapped onto X. Thusthe indeterminacy locus of ν−1 f : Y 99K Xν contains no strata of(Y,BY ). By modifying (Y,BY ) suitably by Proposition 6.3.1, we mayassume that f : Y → X factors through Xν .

Y

f

f

!!CCC

CCCC

C

Xνν

// X

Note that the composition

OX → ν∗OXν → ν∗f∗OY = f∗OY ' OXis an isomorphism. This implies that OX ' ν∗OXν . Therefore, X isnormal.

We introduce Nqklt(X,ω), which is a generalization of the notionof non-klt loci of normal pairs.

6.3. BASIC PROPERTIES OF QUASI-LOG SCHEMES 213

Notation 6.3.6. Let [X,ω] be a quasi-log scheme. The unionof X−∞ with all qlc centers of [X,ω] is denoted by Nqklt(X,ω). IfNqklt(X,ω) 6= X−∞, then

[Nqklt(X,ω), ω|Nqklt(X,ω)]

is a quasi-log scheme by Theorem 6.3.4 (i). Note that Nqklt(X,ω) isdenoted by LCS(X) and is called the LCS locus of a quasi-log scheme[X,ω] in [Am1, Definition 4.6].

Theorem 6.3.7 is also a key result for the theory of quasi-log schemes.

Theorem 6.3.7 (see [Am1, Proposition 4.8] and [F17, Theorem3.45]). Assume that [X,ω] is a quasi-log scheme with X−∞ = ∅. Thenwe have the following properties.

(i) The intersection of two qlc strata is a union of qlc strata.(ii) For any closed point x ∈ X, the set of all qlc strata passing

through x has a unique minimal (with respect to the inclusion)element Cx. Moreover, Cx is normal at x.

Proof. Let C1 and C2 be two qlc strata of [X,ω]. We fix P ∈C1∩C2. It is enough to find a qlc stratum C such that P ∈ C ⊂ C1∩C2.The union X ′ = C1 ∪ C2 with ω′ = ω|X′ is a qlc pair having twoirreducible components. Hence, it is not normal at P . By Lemma6.3.5, P ∈ Nqklt(X ′, ω′). Therefore, there exists a qlc stratum C ⊂ C1

with dimC < dimC1 such that P ∈ C ∩ C2. If C ⊂ C2, then we aredone. Otherwise, we repeat the argument with C1 = C and reach theconclusion in a finite number of steps. So, we finish the proof of (i). Theuniqueness of the minimal (with respect to the inclusion) qlc stratumfollows from (i) and the normality of the minimal stratum follows fromLemma 6.3.5. Thus, we have (ii).

Lemma 6.3.8 is obvious. We will sometimes use it implicitly in thetheory of quasi-log schemes.

Lemma 6.3.8. Let [X,ω] be a quasi-log scheme. Assume that X =V ∪X−∞ and V ∩X−∞ = ∅. Then [V, ω|V ] is a quasi-log scheme withonly quasi-log canonical singularities.

The following lemma is a slight generalization of [F17, Lemma3.71], which will play a crucial role in the proof of the rationality the-orem for quasi-log schemes (see Theorem 6.6.1).

Lemma 6.3.9 (see [F17, Lemma 3.71] and [F40, Lemma 3.16]).Let [X,ω] be a quasi-log scheme with X−∞ = ∅ and let x ∈ X be aclosed point. Let D1, D2, · · · , Dk be effective Cartier divisors on Xsuch that x ∈ SuppDi for every i. Let f : (Y,BY )→ X be a quasi-log

214 6. FUNDAMENTAL THEOREMS FOR QUASI-LOG SCHEMES

resolution. Assume that the normalization of (Y,BY +∑k

i=1 f∗Di) is

sub log canonical. This means that (Y ν ,Ξ) is sub log canonical, whereν : Y ν → Y is the normalization and

KY ν + Ξ = ν∗(KY +BY +k∑i=1

f ∗Di).

Note that it requires that no irreducible component of Y is mapped into∪ki=1 SuppDi. Then k ≤ dimxX. More precisely, k ≤ dimxCx, where

Cx is the minimal qlc stratum of [X,ω] passing through x.

Proof. We prove this lemma by induction on the dimension.

Step 1. By Proposition 6.3.1, we may assume that

(Y, SuppBY +k∑i=1

f∗Di)

is a globally embedded simple normal crossing pair. Note that

f∗OY (d−(B<1Y )e) ' OX .

Therefore, there is a stratum Si of (Y,BY + f∗Di) mapped onto Di

for every i. Note that f : (Y,BY +∑k

i=1 f∗Di) → X gives a natural

quasi-log structure on [X,ω +∑k

i=1Di] with only quasi-log canonicalsingularities.

Step 2. In this step, we assume that dimxX = 1. If x is a qlcstratum of [X,ω], then we have k = 0. Therefore, we may assumethat x is not a qlc stratum of [X,ω]. By shrinking X around x, wemay assume that every stratum of (Y,BY ) is mapped onto X. ThenX is irreducible and normal (see Lemma 6.3.5), and f : Y → X is flat.In this case, we can easily check that f∗OY (d−(B<1

Y )e) ' OX impliesk ≤ 1 = dimxX.

Step 3. We assume that dimxX ≥ 2. If x is a qlc stratum of [X,ω],then k = 0. So we may assume that x is not a qlc stratum of [X,ω]. LetC be the minimal qlc stratum of [X,ω] passing through x. By shrinkingX around x, we may assume that C is normal (see Theorem 6.3.7). ByProposition 6.3.1, we may assume that the union of all strata of (Y,BY )mapped to C, which is denoted by Y ′, is a union of some irreduciblecomponents of Y . Then f : (Y ′, BY ′) → C gives a natural quasi-logstructure induced by the original quasi-log structure f : (Y,BY ) → X(see Theorem 3.2.7). Therefore, by the induction on the dimension,we have k ≤ dimxC ≤ dimxX when dimxC < dimxX. Thus wemay assume that X is the unique qlc stratum of [X,ω]. Note that f :

6.4. ON QUASI-LOG STRUCTURES OF NORMAL PAIRS 215

(Y,BY + f ∗D1)→ X gives a natural quasi-log structure on [X,ω+D1]with only quasi-log canonical singularities. Let X ′ be the union of qlcstrata of [X,ω + D1] contained in SuppD1. Then [X ′, (ω + D1)|X′ ] isa qlc pair with dimxX

′ < dimxX. Note that [X ′, (ω + D1)|X′ ] withD2|X′ , · · · , Dk|X′ satisfies the condition similar to the original one for[X,ω] with D1, · · · , Dk. Therefore, k − 1 ≤ dimxX

′ < dimxX. Thisimplies k ≤ dimxX.

Anyway, we obtained the desired inequality k ≤ dimxCx, where Cxis the minimal qlc stratum of [X,ω] passing through x.

6.4. On quasi-log structures of normal pairs

In this section, we see that a normal pair has a natural quasi-logstructure. By this fact, we can use the theory of quasi-log schemesfor the study of normal pairs. Moreover, we discuss toric varieties andtoric polyhedra as examples of quasi-log schemes.

6.4.1 (Quasi-log structures for normal pairs). Let (X,B) be a nor-mal pair, that is, X is a normal variety and B is an effective R-divisoron X such that KX + B is R-Cartier. We put ω = KX + B. Letf : Y → X be a resolution such that Supp f−1

∗ B ∪ Exc(f) is a simplenormal crossing divisor on Y . We put

KY +BY = f ∗(KX +B).

Since B is effective, d−(B<1Y )e is effective and f -exceptional. Therefore,

f∗OY (d−(B<1Y )e) ' OX . We put

IX−∞ = f∗OY (d−(B<1Y )e − bB>1

Y c).

Then IX−∞ is an ideal sheaf on X. By Proposition 6.3.1, IX−∞ isindependent of the resolution f : Y → X. It is nothing but the non-lcideal sheaf JNLC(X,B) of the pair (X,B). By this ideal sheaf IX−∞ , wedefine a proper closed subscheme X−∞ of X. Let C be the set of logcanonical strata of (X,B). Then, by definition, the set C coincideswith the images of (Y,BY )-strata that are not included in X−∞. Weput M = Y ×C and D = B×C. Then (Y,BY ) ' (Y ×0, BY ×0)is a globally embedded simple normal crossing pair with the ambientspace M . Therefore, the data(

X,ω, f : (Y,BY )→ X)

gives a natural quasi-log structure, which is compatible with the orig-inal structure of the normal pair (X,B). By the above description,(X,B) is log canonical if and only if

(X,ω, f : (Y,BY )→ X

)has only

216 6. FUNDAMENTAL THEOREMS FOR QUASI-LOG SCHEMES

qlc singularities. Therefore, the notion of qlc pairs is a generalizationof that of log canonical pairs.

6.4.2. We use the same notation as in 6.4.1. Let X ′ be the union ofX−∞ with a union of some log canonical centers of (X,B). Then, byTheorem 6.3.4 (i), X ′ has a natural quasi-log structure with ω′ = ω|X′

and X ′−∞ = X−∞. Moreover, the qlc strata of [X ′, ω′] are exactlythe log canonical centers of (X,B) that are contained in X ′. By theproof of Theorem 6.3.4 and Proposition 6.3.1, we can easily check thatthe quasi-log structure of [X ′, ω′] is essentially unique, that is, it isindependent of the resolution f : Y → X.

By the descriptions in 6.4.1 and 6.4.2, Theorem 3.16.4 and Theorem3.16.5 become special cases of Theorem 6.3.4 (ii).

We can treat toric varieties and toric polyhedra as examples ofquasi-log schemes.

6.4.3 (Toric polyhedron). Here, we freely use the basic notation ofthe toric geometry (see, for example, [Ful]).

Definition 6.4.4. For a subset Φ of a fan ∆, we say that Φ is starclosed if σ ∈ Φ, τ ∈ ∆ and σ ≺ τ imply τ ∈ Φ.

Definition 6.4.5 (Toric polyhedron). For a star closed subset Φof a fan ∆, we denote by Y = Y (Φ) the subscheme

∪σ∈Φ V (σ) of

X = X(∆), and we call it the toric polyhedron associated to Φ.

Let X = X(∆) be a toric variety and let D be the complement ofthe big torus. Then the following property is well known and is easyto check.

Proposition 6.4.6. The pair (X,D) is log canonical and KX+D ∼0. Let W be a closed subvariety of X. Then, W is a log canonical centerof (X,D) if and only if W = V (σ) for some σ ∈ ∆ \ 0.

Therefore, we have the following theorem by adjunction (see Theo-rem 6.3.4 (i)).

Theorem 6.4.7. Let Y = Y (Φ) be a toric polyhedron on X =X(∆). Then, the log canonical pair (X,D) induces a natural quasi-logstructure on [Y, 0]. Note that [Y, 0] has only qlc singularities. Let Wbe a closed subvariety of Y . Then, W is a qlc stratum of [Y, 0] if andonly if W = V (σ) for some σ ∈ Φ.

Thus, we can use the theory of quasi-log schemes to investigatetoric varieties and toric polyhedra. For example, we have the followingresult as a special case of Theorem 6.3.4 (ii).

6.5. BASEPOINT-FREE THEOREM FOR QUASI-LOG SCHEMES 217

Corollary 6.4.8. We use the same notation as in Theorem 6.4.7.Assume that X is projective and L is an ample Cartier divisor on X.Then H i(X, IY ⊗OX(L)) = 0 for every i > 0, where IY is the definingideal sheaf of Y on X. In particular, the restriction map

H0(X,OX(L))→ H0(Y,OY (L))

is surjective.

We can prove various vanishing theorems for toric varieties and toricpolyhedra without appealing the results in Chapter 5. For the details,see [F11] and [F16].

6.5. Basepoint-free theorem for quasi-log schemes

Theorem 6.5.1 is the main theorem of this section. It is [Am1,Theorem 5.1].

Theorem 6.5.1 (Basepoint-free theorem). Let [X,ω] be a quasi-logscheme and let π : X → S be a projective morphism between schemes.Let L be a π-nef Cartier divisor on X. Assume that

(i) qL− ω is π-ample for some real number q > 0, and(ii) OX−∞(mL) is π|X−∞-generated for every m 0.

Then OX(mL) is π-generated for every m 0, that is, there exists apositive number m0 such that OX(mL) is π-generated for every m ≥m0.

Before we prove Theorem 6.5.1, let us prepare some lemmas. Lemma6.5.2 is a variant of Shokurov’s concentration method (see, for example,[Sh1] and [KoMo, Section 3.5]).

Lemma 6.5.2 (see [F28, Lemma 12.2]). Let f : Y → Z be a projec-tive morphism from a normal variety Y onto an affine variety Z. Let Vbe a general closed subvariety of Y such that f : V → Z is genericallyfinite. Let M be an f -ample R-divisor on Y . Assume that

(M |F )d > kmd,

where F is a general fiber of f : Y → Z, d = dimF , and k is themapping degree of f : V → Z. Then we can find an effective R-CartierR-divisor D on Y such that

D ∼R M

and that multVD > m. If M is a Q-divisor, then we can make D aQ-divisor with D ∼Q M .

218 6. FUNDAMENTAL THEOREMS FOR QUASI-LOG SCHEMES

Proof. We can write

M = M1 + a2M2 + · · ·+ alMl,

where M1 is an f -ample Q-divisor such that (M1|F )d > kmd, ai is apositive real number, and Mi is an f -ample Cartier divisor for every i.If M is a Q-divisor, then we may assume that l = 2 and a2 is rational.Let IV be the defining ideal sheaf of V on Y . We consider the followingexact sequence

0→ f∗(OY (pM1)⊗ IpmV )→ f∗OY (pM1)

→ f∗(OY (pM1)⊗OY /IpmV )→ · · ·

for a sufficiently large and divisible positive integer p. By restrictingthe above sequence to a sufficiently general fiber F of f , we can checkthat the rank of f∗OY (pM1) is greater than that of f∗(OY (pM1) ⊗OY /IpmV ) by the usual estimates (see Lemma 6.5.3 below). Therefore,f∗(OY (pM1)⊗ IpmV ) 6= 0. Let D1 be a member of

H0(Z, f∗(OY (pM1)⊗ IpmV )) = H0(Y,OY (pM1)⊗ IpmV )

and let Di be an effective Q-Cartier Q-divisor such that Di ∼Q Mi fori ≥ 2. We can take D2 with multVD2 > 0. Then D = (1/p)D1 +a2D2 + · · ·+ alDl satisfies the desired properties.

We note the following well-known lemma. The proof is obvious.

Lemma 6.5.3. Let X be a normal projective variety with dimX = dand let A be an ample Q-divisor on X such that aA is Cartier for somepositive integer a. Then

h0(X,OX(taA)) = χ(X,OX(taA))

=(taA)d

d!+ (lower terms in t)

by the Riemann–Roch formula and Serre’s vanishing theorem for everyt 0.

Let P ∈ X be a smooth point. Then

dimCOX/mαP =

(α− 1 + d

d

)=αd

d!+ (lower terms in α)

for all α ≥ 1, where mP is the maximal ideal associated to P .

Let us start the proof of Theorem 6.5.1.

6.5. BASEPOINT-FREE THEOREM FOR QUASI-LOG SCHEMES 219

Proof of Theorem 6.5.1. Without loss of generality, we mayassume that S is affine. We use induction on the dimension of dimX \X−∞. Theorem 6.5.1 is obviously true when dimX \X−∞ = 0.

Claim 1. OX(mL) is π-generated around Nqklt(X,ω) for everym 0.

Proof of Claim. We putX ′ = Nqklt(X,ω). Then [X ′, ω′], whereω′ = ω|X′ , is a quasi-log scheme by adjunction when X ′ 6= X−∞ (seeTheorem 6.3.4 (i)). If X ′ = X−∞, then OX′(mL) is π-generated forevery m 0 by the assumption (ii). If X ′ 6= X−∞, then OX′(mL)is π-generated for every m 0 by induction on the dimension ofX \X−∞. By Theorem 6.3.4 (ii), R1π∗(IX′ ⊗OX(mL)) = 0 for everym ≥ q. Therefore, the restriction map π∗OX(mL)) → π∗OX′(mL) issurjective for every m ≥ q. By the following commutative diagram:

π∗π∗OX(mL)

α // π∗π∗OX′(mL)

// 0

OX(mL) // OX′(mL) // 0,

we know that OX(mL) is π-generated around X ′ for every m 0. Claim 2. OX(mL) is π-generated on a non-empty Zariski open set

for every m 0.

Proof of Claim. By Claim 1, we may assume that Nqklt(X,ω)is empty. Without loss of generality, we may assume that X is con-nected. Then, by Theorem 6.3.7 and Lemma 6.3.5, X is irreducibleand normal. Therefore, we may further assume that S is an irreduciblevariety.

If L is π-numerically trivial, then π∗OX(L) is not zero. This isbecause

h0(Xη,OXη(L)) = χ(Xη,OXη(L))

= χ(Xη,OXη) = h0(Xη,OXη) > 0

by Theorem 6.3.4 (ii) and by [Kle, Chapter II §2 Theorem 1], whereXη is the generic fiber of π : X → S. Let D be a general member of|L|. Let f : (Y,BY ) → X be a quasi-log resolution and let M be theambient space of (Y,BY ). By taking blow-ups of M , we may assumethat (Y, SuppBY +f ∗D) is a globally embedded simple normal crossingpair by Proposition 6.3.1. We note that every stratum of (Y,BY ) ismapped onto X by assumption. We can take a positive real numberc ≤ 1 such that BY + cf ∗D is a subboundary and some stratum of

220 6. FUNDAMENTAL THEOREMS FOR QUASI-LOG SCHEMES

(Y,BY + cf ∗D) does not dominate X. Note that f∗OY (d−(B<1Y )e) '

OX . Then f : (Y,BY + cf ∗D)→ X gives a natural quasi-log structureon the pair [X,ω + cD] with only qlc singularities. We note that qL−(ω + cD) is π-ample. By Claim 1, OX(mL) is π-generated aroundNqklt(X,ω + cD) 6= ∅ for every m 0. So, we may assume that L isnot π-numerically trivial.

We take a general closed subvariety V of X such that π : V → Sis generically finite. Then we can take an effective R-Cartier R-divisorD on X such that multVD > k · codimXV , where k is the mappingdegree of π : V → S, and that D ∼R (q + r)L − ω for some r >0 by Lemma 6.5.2. By taking blow-ups of M , we may assume that(Y, SuppBY + f∗D) is a globally embedded simple normal crossingpair by Proposition 6.3.1. By the construction of D, we can find apositive real number c < 1 such that BY + cf ∗D is a subboundaryand some stratum of (Y,BY + cf ∗D) does not dominate X. Notethat f∗OY (d−(B<1

Y )e) ' OX . Then f : (Y,BY + cf ∗D) → X givesa natural quasi-log structure on the pair [X,ω + cD] with only qlcsingularities. We note that q′L− (ω + cD) is π-ample by c < 1, whereq′ = q+cr. By construction, Nqklt(X,ω+cD) is non-empty. Therefore,by applying Claim 1 to [X,ω + cD], OX(mL) is π-generated aroundNqklt(X,ω + cD) for every m 0.

So, we finish the proof of Claim 2.

Let p be a prime number and let l be a large integer. Then wehave that π∗OX(plL) 6= 0 by Claim 2 and that OX(plL) is π-generatedaround Nqklt(X,ω) by Claim 1.

Claim 3. If the relative base locus Bsπ |plL| (with the reduced schemestructure) is not empty, then there exists a positive integer a such thatBsπ |palL| is strictly smaller than Bsπ |plL|.

Proof of Claim. Let f : (Y,BY ) → X be a quasi-log resolu-tion. We take a general member D ∈ |plL|. We note that S isaffine and |plL| is free around Nqklt(X,ω). Thus, f ∗D intersects anystrata of (Y, SuppBY ) transversally over X \ Bsπ |plL| by Bertini andf ∗D contains no strata of (Y,BY ). By taking blow-ups of M suit-ably, we may assume that (Y, SuppBY + f∗D) is a global embeddedsimple normal crossing pair by Proposition 6.3.1. We take the max-imal positive real number c such that BY + cf ∗D is a subboundaryover X \ X−∞. We note that c ≤ 1. Here, we used the fact thatOX ' f∗OY (d−(B<1

Y )e) over X \X−∞. Then f : (Y,BY + cf ∗D)→ Xgives a natural quasi-log structure on the pair [X,ω′ = ω + cD]. Notethat Nqlc(X,ω) = Nqlc(X,ω′) by construction. We also note that

6.6. RATIONALITY THEOREM FOR QUASI-LOG SCHEMES 221

[X,ω′] has a qlc center C that intersects Bsπ |plL| by construction.By induction on the dimension, OC∪X−∞(mL) is π-generated for everym 0 since (q+ cpl)L− (ω+ cD) is π-ample. We can lift the sectionsof OC∪X−∞(mL) to X for m ≥ q + cpl by Theorem 6.3.4 (ii). Thenwe obtain that OX(mL) is π-generated around C for every m 0.Therefore, Bsπ |palL| is strictly smaller than Bsπ |plL| for some positiveinteger a

Claim 4. OX(mL) is π-generated for every m 0.

Proof of Claim. By Claim 3 and the noetherian induction, we

obtain that OX(plL) and OX(p′l′L) are π-generated for large l and l′,

where p and p′ are prime numbers and they are relatively prime. So,there exists a positive number m0 such that OX(mL) is π-generatedfor every m ≥ m0.

Thus we obtained the desired result.

Corollary 6.5.4 is a special case of Theorem 6.5.1. Note that a logcanonical pair has a natural quasi-log structure with only qlc singular-ities (see 6.4.1).

Corollary 6.5.4 (Basepoint-free theorem for log canonical pairs,see [F27] and [F28]). Let (X,∆) be a log canonical pair and let π :X → S be a projective morphism. Let L be a π-nef Cartier divisoron X. Assume that qL − (KX + ∆) is π-ample for some positive realnumber q. Then OX(mL) is π-generated for every m 0.

We strongly recommend the reader to see [F28, Section 2], where wedescribe the difference between the approach discussed in this sectionand the framework established in [F28].

6.6. Rationality theorem for quasi-log schemes

In this section, we prove the following rationality theorem for quasi-log schemes (see [Am1, Theorem 5.9]).

Theorem 6.6.1 (Rationality theorem). Assume that [X,ω] is aquasi-log scheme such that ω is Q-Cartier. This means that ω is R-linearly equivalent to a Q-Cartier divisor (or a Q-line bundle) on X.Let π : X → S be a projective morphism between schemes and let H bea π-ample Cartier divisor on X. Assume that r is a positive numbersuch that

(1) H + rω is π-nef but not π-ample, and(2) (H + rω)|X−∞ is π|X−∞-ample.

222 6. FUNDAMENTAL THEOREMS FOR QUASI-LOG SCHEMES

Then r is a rational number, and in reduced form, r has denominatorat most a(dimX + 1), where aω is R-linearly equivalent to a Cartierdivisor (or a line bundle) on X.

Before the proof of Theorem 6.6.1, we recall the following lemmas.

Lemma 6.6.2 (see [KoMo, Lemma 3.19]). Let P (x, y) be a non-trivial polynomial of degree ≤ n and assume that P vanishes for allsufficiently large integral solutions of 0 < ay − rx < ε for some fixedpositive integer a and positive ε for some r ∈ R. Then r is rational,and in reduced form, r has denominator ≤ a(n+ 1)/ε.

Proof. We assume that r is irrational. Then an infinite number ofintegral points in the (x, y)-plane on each side of the line ay−rx = 0 arecloser than ε/(n+ 2) to that line. So there is a large integral solution(x′, y′) with 0 < ay′ − rx′ < ε/(n+ 2). In this case, we see that

(2x′, 2y′), · · · , ((n+ 1)x′, (n+ 1)y′)

are also solutions by hypothesis. So (y′x− x′y) divides P , since P and(y′x − x′y) have (n + 1) common zeroes. We choose a smaller ε andrepeat the argument. We do this n+ 1 times to get a contradiction.

Now we assume that r = u/v in lowest terms. For given j, let(x′, y′) be a solution of ay − rx = aj/v. Note that an integral solutionexists for every j. Then we have a(y′+ku)−r(x′+akv) = aj/v for all k.So, as above, if aj/v < ε, (ay− rx)− (aj/v) must divide P . Therefore,we can have at most n such values of j. Thus a(n+ 1)/v ≥ ε.

Lemma 6.6.3. Let C be a projective variety and let D1 and D2 beCartier divisors on X. Consider the Hilbert polynomial

P (u1, u2) = χ(C,OC(u1D1 + u2D2)).

If D1 is ample, then P (u1, u2) is a nontrivial polynomial of total degree≤ dimC. This is because P (u1, 0) = h0(C,OC(u1D1)) 6≡ 0 if u1 issufficiently large.

Proof of Theorem 6.6.1. Without loss of generality, we mayassume that aω itself is Cartier. Let m be a positive integer such thatH ′ = mH is π-very ample. If H ′ + r′ω is π-nef but not π-ample, and(H ′ + r′ω)|Nqlc(X,ω) is π|Nqlc(X,ω)-ample, then we have

H + rω =1

m(H ′ + r′ω).

This gives r = 1mr′. Thus, r is rational if and only if r′ is rational. As-

sume furthermore that r′ has denominator v. Then r has denominatordividing mv. Since m can be arbitrary sufficiently large integer, this

6.6. RATIONALITY THEOREM FOR QUASI-LOG SCHEMES 223

implies that r has denominator dividing v. Therefore, by replacing Hwith mH, we may assume that H is very ample over S.

For each (p, q) ∈ Z2, let L(p, q) denote the relative base locus ofthe linear system associated to M(p, q) = pH + qaω on X (with thereduced scheme structure), that is,

L(p, q) = Supp(Coker(π∗π∗OX(M(p, q))→ OX(M(p, q)))).

By definition, L(p, q) = X if and only if π∗OX(M(p, q)) = 0.

Claim 1 (cf. [KoMo, Claim 3.20]). Let ε be a positive real numberwith ε ≤ 1. For (p, q) sufficiently large and 0 < aq − rp < ε, L(p, q) isthe same subset of X. We call this subset L0. Let I ⊂ Z2 be the set of(p, q) for which 0 < aq− rp < 1 and L(p, q) = L0. Then I contains allsufficiently large (p, q) with 0 < aq − rp < 1.

Proof. We fix (p0, q0) ∈ Z2 such that p0 > 0 and 0 < aq0−rp0 < 1.Since H is π-very ample, there exists a positive integer m0 such thatOX(mH + jaω) is π-generated for every m > m0 and every 0 ≤ j ≤q0 − 1. Let M be the round-up of(

m0 +1

r

)/(ar− p0

q0

).

If (p′, q′) ∈ Z2 such that 0 < aq′ − rp′ < 1 and q′ ≥ M + q0 − 1, thenwe can write

p′H + q′aω = k(p0H + q0aω) + (lH + jaω)

for some k ≥ 0, 0 ≤ j ≤ q0 − 1 with l > m0. This is because wecan uniquely write q′ = kq0 + j with 0 ≤ j ≤ q0 − 1. Thus, we havekq0 ≥M . So, we obtain

l = p′ − kp0 >a

rq′ − 1

r− (kq0)

p0

q0≥

(ar− p0

q0

)M − 1

r≥ m0.

Therefore, L(p′, q′) ⊂ L(p0, q0). By the noetherian induction, we obtainthe desired closed subset L0 ⊂ X and I ⊂ Z2.

Claim 2. We have L0 ∩ Nqlc(X,ω) = ∅.

Proof. We take (α, β) ∈ Q2 such that α > 0, β > 0, and βa/α > ris sufficiently close to r. Then (αH + βaω)|Nqlc(X,ω) is π|Nqlc(X,ω)-amplebecause (H + rω)|Nqlc(X,ω) is π|Nqlc(X,ω)-ample. If 0 < aq − rp < 1 and(p, q) ∈ Z2 is sufficiently large, then

M(p, q) = mM(α, β) + (M(p, q)−mM(α, β))

such that M(p, q)−mM(α, β) is π-very ample and that

m(αH + βaω)|Nqlc(X,ω)

224 6. FUNDAMENTAL THEOREMS FOR QUASI-LOG SCHEMES

is also π|Nqlc(X,ω)-very ample. It can be checked by the same argumentas in the proof of Claim 1. Therefore, ONqlc(X,ω)(M(p, q)) is π-veryample. Since

π∗OX(M(p, q))→ π∗ONlc(X,B)(M(p, q))

is surjective by the vanishing theorem: Theorem 6.3.4 (ii), we obtainL(p, q) ∩ Nqlc(X,ω) = ∅. We note that

M(p, q)− ω = pH + (qa− 1)ω

is π-ample because (p, q) is sufficiently large and aq−rp < 1. By Claim1, we have L0 ∩ Nqlc(X,ω) = ∅.

Claim 3. We assume that r is not rational or that r is rationaland has denominator > a(n + 1) in reduced form, where n = dimX.Then, for (p, q) sufficiently large and 0 < aq − rp < 1, OX(M(p, q)) isπ-generated at the generic point of every qlc stratum of [X,ω].

Proof of Claim. We note that

M(p, q)− ω = pH + (qa− 1)ω.

If aq − rp < 1 and (p, q) is sufficiently large, then M(p, q) − ω is π-ample. Let C be a qlc stratum of [X,ω]. We note that we may assumeC ∩ Nqlc(X,ω) = ∅ by Claim 2. Then

PCη(p, q) = χ(Cη,OCη(M(p, q)))

is a non-zero polynomial of degree at most dimCη ≤ dimX by Lemma6.6.3. Note that Cη is the generic fiber of C → π(C). By Lemma 6.6.2,there exists (p, q) such that PCη(p, q) 6= 0, (p, q) sufficiently large, and0 < aq − rp < 1. By the π-ampleness of M(p, q)− ω,

PCη(p, q) = χ(Cη,OCη(M(p, q))) = h0(Cη,OCη(M(p, q)))

and

π∗OX(M(p, q))→ π∗OC(M(p, q))

is surjective by Theorem 6.3.4 (ii). We note that

R1π∗(IC∪Nqlc(X,ω) ⊗OX(M(p, q))) = 0

by the vanishing theorem (see Theorem 6.3.4 (ii)) and that Nqlc(X,ω)∩C = ∅. Therefore, OX(M(p, q)) is π-generated at the generic point ofC. By combining this with Claim 1, OX(M(p, q)) is π-generated at thegeneric point of every qlc stratum of [X,ω] if (p, q) is sufficiently largewith 0 < aq − rp < 1. So, we obtain Claim 3.

6.6. RATIONALITY THEOREM FOR QUASI-LOG SCHEMES 225

Note that OX(M(p, q)) is not π-generated for (p, q) ∈ I becauseM(p, q) is not π-nef. Therefore, L0 6= ∅. We shrink S to an affine opensubset intersecting π(L0). Let D1, · · · , Dn+1 be general members of

π∗OX(M(p0, q0)) = H0(X,OX(M(p0, q0)))

with (p0, q0) ∈ I. Let f : (Y,BY ) → X be a quasi-log resolution of[X,ω]. We consider f : (Y,BY +

∑n+1i=1 f

∗Di)→ X. By taking blow-upsof the ambient space M of (Y,BY ), we may assume that (Y, SuppBY +∑n+1

i=1 f∗Di) is a globally embedded simple normal crossing pair by

Proposition 6.3.1. We can take the maximal positive real number csuch that BY + c

∑n+1i=1 f

∗Di is subboundary over X \ Nqlc(X,ω) byClaim 3. Note that f ∗Di contains no strata of (Y,BY ) for every isince Di is a general member of H0(X,OX(M(p, q))) for every i (seeClaim 3). On the other hand, c < 1 by Lemma 6.3.9. Then f :(Y,BY + c

∑n+1i=1 f

∗Di) → X gives a natural quasi-log structure on

the pair [X,ω + cD], where D =∑n+1

i=1 Di. By construction, the pair[X,ω + cD] has some qlc centers contained in L0. Let C be a qlccenter contained in L0. We note that Nqlc(X,ω) = Nqlc(X,ω+cD) byconstruction. In particular, C∩Nqlc(X,ω+cD) = C∩Nqlc(X,ω) = ∅.We consider

ω + cD = c(n+ 1)p0H + (1 + c(n+ 1)q0a)ω.

Thus we have

pH + qaω − (ω + cD)

= (p− c(n+ 1)p0)H + (qa− (1 + c(n+ 1)q0a))ω.

If p and q are large enough and 0 < aq − rp ≤ aq0 − rp0, then

pH + qaω − (ω + cD)

is π-ample. This is because

(p− c(n+ 1)p0)H + (qa− (1 + c(n+ 1)q0a))ω

= (p− (1 + c(n+ 1))p0)H + (qa− (1 + c(n+ 1))q0a)ω

+ p0H + (q0a− 1)ω.

Suppose that r is not rational. There must be arbitrarily large (p, q)such that 0 < aq − rp < ε = aq0 − rp0 and

χ(Cη,OCη(M(p, q))) 6= 0

by Lemma 6.6.2 because

PCη(p, q) = χ(Cη,OCη(M(p, q)))

226 6. FUNDAMENTAL THEOREMS FOR QUASI-LOG SCHEMES

is a nontrivial polynomial of degree at most dimCη by Lemma 6.6.3.Since M(p, q) − (ω + cD) is π-ample by 0 < aq − rp < aq0 − rp0, wehave

h0(Cη,OCη(M(p, q))) = χ(Cη,OCη(M(p, q))) 6= 0

by the vanishing theorem: Theorem 6.3.4 (ii). By the vanishing theo-rem: Theorem 6.3.4 (ii),

π∗OX(M(p, q))→ π∗OC(M(p, q))

is surjective because M(p, q)− (ω + cD) is π-ample. We note that

R1π∗(IC∪Nqlc(X,ω+cD) ⊗OX(M(p, q))) = 0

by Theorem 6.3.4 (ii) and that C ∩ Nqlc(X,ω + cD) = ∅. Thus Cis not contained in L(p, q). Therefore, L(p, q) is a proper subset ofL(p0, q0) = L0, giving the desired contradiction. So now we know thatr is rational.

We next suppose that the assertion of the theorem concerning thedenominator of r is false. We choose (p0, q0) ∈ I such that aq0 − rp0 isthe maximum, say it is equal to d/v. If 0 < aq− rp ≤ d/v and (p, q) issufficiently large, then

χ(Cη,OCη(M(p, q))) = h0(Cη,OCη(M(p, q)))

since M(p, q) − (ω + cD) is π-ample. There exists sufficiently large(p, q) in the strip 0 < aq − rp < 1 with ε = 1 for which

h0(Cη,OCη(M(p, q))) = χ(Cη,OCη(M(p, q))) 6= 0

by Lemma 6.6.2 since χ(Cη,OCη(M(p, q))) is a nontrivial polynomialof degree at most dimCη by Lemma 6.6.3. Note that aq − rp ≤ d/v =aq0 − rp0 holds automatically for (p, q) ∈ I. Since

π∗OX(M(p, q))→ π∗OC(M(p, q))

is surjective by the π-ampleness of M(p, q)− (ω + cD), we obtain thedesired contradiction by the same reason as above. So, we finish theproof of the rationality theorem.

6.7. Cone theorem for quasi-log schemes

The main theorem of this section is the cone theorem for quasi-log schemes (see [Am1, Theorem 5.10]). Before we state the maintheorem, let us fix the notation.

Definition 6.7.1 (see [Am1, Definition 5.2]). Let [X,ω] be aquasi-log scheme with the non-qlc locus X−∞. Let π : X → S bea projective morphism between schemes. We put

NE(X/S)−∞ = Im(NE(X−∞/S)→ NE(X/S)).

6.7. CONE THEOREM FOR QUASI-LOG SCHEMES 227

We sometimes use NE(X/S)Nqlc(X,ω) to denote NE(X/S)−∞. For anR-Cartier divisor D, we define

D≥0 = z ∈ N1(X/S) | D · z ≥ 0.

Similarly, we can define D>0, D≤0, and D<0. We also define

D⊥ = z ∈ N1(X/S) | D · z = 0.

We use the following notation

NE(X/S)D≥0 = NE(X/S) ∩D≥0,

and similarly for > 0, ≤ 0, and < 0.

Definition 6.7.2 (see [Am1, Definition 5.3]). An extremal face ofNE(X/S) is a non-zero subcone F ⊂ NE(X/S) such that z, z′ ∈ Fand z+z′ ∈ F imply that z, z′ ∈ F . Equivalently, F = NE(X/S)∩H⊥for some π-nef R-divisor H, which is called a supporting function of F .An extremal ray is a one-dimensional extremal face.

(1) An extremal face F is called ω-negative if F ∩NE(X/S)ω≥0 =0.

(2) An extremal face F is called rational if we can choose a π-nefQ-divisor H as a support function of F .

(3) An extremal face F is called relatively ample at infinity if F ∩NE(X/S)−∞ = 0. Equivalently, H|X−∞ is π|X−∞-ample forany supporting function H of F .

(4) An extremal face F is called contractible at infinity if it hasa rational supporting function H such that H|X−∞ is π|X−∞-semi-ample.

The following theorem is a direct consequence of Theorem 6.5.1.

Theorem 6.7.3 (Contraction theorem). Let [X,ω] be a quasi-logscheme and let π : X → S be a projective morphism between schemes.Let H be a π-nef Cartier divisor such that F = H⊥ ∩ NE(X/S) isω-negative and contractible at infinity. Then there exists a projectivemorphism ϕF : X → Y over S with the following properties.

(1) Let C be an integral curve on X such that π(C) is a point.Then ϕF (C) is a point if and only if [C] ∈ F .

(2) OY ' (ϕF )∗OX .(3) Let L be a line bundle on X such that L ·C = 0 for every curve

C with [C] ∈ F . Assume that L⊗m|X−∞ is ϕF |X−∞-generatedfor every m 0. Then there is a line bundle LY on Y suchthat L ' ϕ∗FLY .

228 6. FUNDAMENTAL THEOREMS FOR QUASI-LOG SCHEMES

Proof. By assumption, qH−ω is π-ample for some positive integerq and H|X−∞ is π|X−∞-semi-ample. By Theorem 6.5.1, OX(mH) is π-generated for some large m. We take the Stein factorization of theassociated morphism. Then, we have the contraction morphism ϕF :X → Y with the properties (1) and (2).

We consider ϕF : X → Y and NE(X/Y ). Then NE(X/Y ) = F ,L is numerically trivial over Y , and −ω is ϕF -ample. Applying thebasepoint-free theorem (see Theorem 6.5.1) over Y , both L⊗m andL⊗(m+1) are pull-backs of line bundles on Y . Their difference gives aline bundle LY such that L ' ϕ∗FLY .

One of the main purposes of this book is to establish the followingtheorem.

Theorem 6.7.4 (Cone theorem). Let [X,ω] be a quasi-log schemeand let π : X → S be a projective morphism between schemes. Thenwe have the following properties.

(1) NE(X/S) = NE(X/S)ω≥0+NE(X/S)−∞+∑Rj, where Rj’s

are the ω-negative extremal rays of NE(X/S) that are ratio-nal and relatively ample at infinity. In particular, each Rj isspanned by an integral curve Cj on X such that π(Cj) is apoint.

(2) Let H be a π-ample R-divisor on X. Then there are onlyfinitely many Rj’s included in (ω + H)<0. In particular, theRj’s are discrete in the half-space ω<0.

(3) Let F be an ω-negative extremal face of NE(X/S) that is rel-atively ample at infinity. Then F is a rational face. In partic-ular, F is contractible at infinity.

We closely follow the proof of the cone theorem in [KMM].

Proof. First, we assume that ω is Q-Cartier. This means that ωis R-linearly equivalent to a Q-Cartier divisor. We may assume thatdimRN1(X/S) ≥ 2 and ω is not π-nef. Otherwise, the theorem isobvious.

Step 1. In this step, we prove:

Claim. We have

NE(X/S) = NE(X/S)ω≥0 +NE(X/S)−∞ +∑F

F,

where F ’s vary among all rational proper ω-negative extremal faces thatare relatively ample at infinity and — denotes the closure with respectto the real topology.

6.7. CONE THEOREM FOR QUASI-LOG SCHEMES 229

Proof of Claim. We put

B = NE(X/S)ω≥0 +NE(X/S)−∞ +∑F

F.

It is clear that NE(X/S) ⊃ B. We note that each F is spanned bycurves on X mapped to points on S by Theorem 6.7.3 (1). SupposingNE(X/S) 6= B, we shall derive a contradiction. There is a separatingfunction M which is Cartier and is not a multiple of ω in N1(X/S)such that M > 0 on B \ 0 and M · z0 < 0 for some z0 ∈ NE(X/S).Let C be the dual cone of NE(X/S)ω≥0, that is,

C = D ∈ N1(X/S) | D · z ≥ 0 for z ∈ NE(X/S)ω≥0.Then C is generated by π-nef divisors and ω. Since M is positive onNE(X/S)ω≥0 \ 0, M is in the interior of C, and hence there exists aπ-ample Q-divisor A such that M − A = L′ + pω in N1(X/S), whereL′ is a π-nef Q-divisor on X and p is a non-negative rational number.Therefore, M is expressed in the form M = H + pω in N1(X/S),where H = A + L′ is a π-ample Q-divisor. The rationality theorem(see Theorem 6.6.1) implies that there exists a positive rational numberr < p such that L = H + rω is π-nef but not π-ample, and L|X−∞ isπ|X−∞-ample. Note that L 6= 0 in N1(X/S), since M is not a multipleof ω. Thus the extremal face FL associated to the supporting functionL is contained in B, which implies M > 0 on FL. Therefore, p < r.This is a contradiction. This completes the proof of Claim.

Step 2. In this step, we prove:

Claim. In the equality of Step 1, it is sufficient to assume thatF ’s vary among all rational ω-negative extremal rays that are relativelyample at infinity.

Proof of Claim. Let F be a rational proper ω-negative extremalface that is relatively ample at infinity, and assume that dimF ≥ 2.Let ϕF : X → W be the associated contraction. Note that −ω isϕF -ample. By Step 1, we obtain

F = NE(X/W ) =∑G

G,

where the G’s are the rational proper ω-negative extremal faces ofNE(X/W ). We note that NE(X/W )−∞ = 0 because ϕF embeds X−∞into W . The G’s are also ω-negative extremal faces of NE(X/S) thatare ample at infinity, and dimG < dimF . By induction, we obtain

(♥) NE(X/S) = NE(X/S)ω≥0 +NE(X/S)−∞ +∑

Rj,

230 6. FUNDAMENTAL THEOREMS FOR QUASI-LOG SCHEMES

where the Rj’s are ω-negative rational extremal rays. Note that eachRj does not intersect NE(X/S)−∞.

Step 3. The contraction theorem (see Theorem 6.7.3) guaranteesthat for each extremal ray Rj there exists a reduced irreducible curveCj on X such that [Cj] ∈ Rj. Let ψj : X → Wj be the contractionmorphism of Rj, and let A be a π-ample Cartier divisor. We set

rj = −A · Cjω · Cj

.

Then A+rjω is ψj-nef but not ψj-ample, and (A+rjω)|X−∞ is ψj|X−∞-ample. By the rationality theorem (see Theorem 6.6.1), expressingrj = uj/vj with uj, vj ∈ Z>0 and (uj, vj) = 1, we have the inequalityvj ≤ a(dimX + 1).

Step 4. Now take π-ample Cartier divisors H1, H2, · · · , Hρ−1 suchthat ω and theHi’s form a basis ofN1(X/S), where ρ = dimRN

1(X/S).By Step 3, the intersection of the extremal rays Rj with the hyperplane

z ∈ N1(X/S) | aω · z = −1in N1(X/S) lie on the lattice

Λ = z ∈ N1(X/S) | aω · z = −1, Hi · z ∈ (a(a(dimX + 1))!)−1Z.This implies that the extremal rays are discrete in the half space

z ∈ N1(X/S) | ω · z < 0.Thus we can omit the closure sign —– from the formula (♥) and thiscompletes the proof of (1) when ω is Q-Cartier.

Step 5. Let H be a π-ample R-divisor on X. We choose 0 < εi 1for 1 ≤ i ≤ ρ− 1 such that H −

∑ρ−1i=1 εiHi is π-ample. Then the Rj’s

included in (ω+H)<0 correspond to some elements of the above latticeΛ for which

∑ρ−1i=1 εiHi · z < 1/a. Therefore, we obtain (2).

Step 6. The vector space V = F⊥ ⊂ N1(X/S) is defined over Qbecause F is generated by some of the Rj’s. There exists a π-ampleR-divisor H such that F is contained in (ω + H)<0. Let 〈F 〉 be thevector space spanned by F . We put

WF = NE(X/S)ω+H≥0 +NE(X/S)−∞ +∑Rj 6⊂F

Rj.

Then WF is a closed cone, NE(X/S) = WF +F , and WF ∩〈F 〉 = 0.The supporting functions of F are the elements of V that are positive onWF \ 0. This is a non-empty open set and thus it contains a rational

6.7. CONE THEOREM FOR QUASI-LOG SCHEMES 231

element that, after scaling, gives a π-nef Cartier divisor L such thatF = L⊥ ∩NE(X/S). Therefore, F is rational. So, we have (3).

From now on, we assume that ω is R-Cartier.

Step 7. Let H be a π-ample R-divisor on X. We shall prove(2). We assume that there are infinitely many Rj’s in (ω +H)<0 andget a contradiction. There exists an affine open subset U of S suchthat NE(π−1(U)/U) has infinitely many (ω + H)-negative extremalrays. So, we shrink S and may assume that S is affine. We can writeH = E+H ′, where H ′ is π-ample, [X,ω+E] is a quasi-log pair with thesame qlc centers and non-qlc locus as [X,ω], and ω + E is Q-Cartier.Since ω +H = ω + E +H ′, we have

NE(X/S) = NE(X/S)ω+H≥0 +NE(X/S)−∞ +∑finite

Rj.

This is a contradiction. Thus, we obtain (2). The statement (1) isa direct consequence of (2). Of course, (3) holds by Step 6 once weobtain (1) and (2).

So, we finish the proof of the cone theorem. By combining Theorem 6.7.4 and the main result of [F33] (see

Theorem 4.11.9), we obtain the cone and contraction theorem for semilog canonical pairs in full generality. For the details, see [F33].

We close this section with the following nontrivial example.

Example 6.7.5 ([F17, Example 3.76]). We consider the first pro-jection p : P1×P1 → P1. We take a blow-up µ : Z → P1×P1 at (0,∞).Let A∞ (resp. A0) be the strict transform of P1×∞ (resp. P1×0)on Z. We define M = PZ(OZ ⊕ OZ(A0)) and X is the restriction ofM on (p µ)−1(0). Then X is a simple normal crossing divisor onM . More explicitly, X is a P1-bundle over (p µ)−1(0) and is obtainedby gluing X1 = P1 × P1 and X2 = PP1(OP1 ⊕ OP1(1)) along a fiber.In particular, (X, 0) is a semi log canonical surface. By construction,M → Z has two sections. Let D+ (resp. D−) be the restriction ofthe section of M → Z corresponding to OZ ⊕OZ(A0)→ OZ(A0)→ 0(resp. OZ⊕OZ(A0)→ OZ → 0). Then it is easy to see that D+ is a nefCartier divisor on X and that the linear system |mD+| is free for everym > 0. Note that M is a projective toric variety. Let E be the sectionof M → Z corresponding to OZ ⊕OZ(A0)→ OZ(A0)→ 0. Then, it iseasy to see that E is a nef Cartier divisor on M . Therefore, the linearsystem |E| is free. In particular, |D+| is free on X since D+ = E|X . So,|mD+| is free for every m > 0. We take a general member B0 ∈ |mD+|with m ≥ 2. We consider KX +B with B = D−+B0 +B1 +B2, where

232 6. FUNDAMENTAL THEOREMS FOR QUASI-LOG SCHEMES

B1 and B2 are general fibers of X1 = P1 × P1 ⊂ X. We note thatB0 does not intersect D−. Then (X,B) is an embedded simple normalcrossing pair. In particular, (X,B) is a semi log canonical surface. Ofcourse, [X,KX +B] has a natural quasi-log structure with only qlc sin-gularities (see also Theorem 4.11.9). It is easy to see that there existsonly one integral curve C on X2 = PP1(OP1 ⊕ OP1(1)) ⊂ X such thatC · (KX + B) < 0. Note that C is nothing but the negative sectionof X2 = PP1(OP1 ⊕ OP1(1)) → P1. We also note that (KX + B)|X1 isample on X1. By the cone theorem (see Theorem 6.7.4), we obtain

NE(X) = NE(X)(KX+B)≥0 + R≥0[C].

By Theorem 6.7.4, we have ϕ : X → W which contracts C. We caneasily see that KW+BW , where BW = ϕ∗B, is not Q-Cartier because Cis not Q-Cartier on X. Therefore, we can not always run the minimalmodel program for semi log canonical surfaces.

The above example implies that the cone and contraction theo-rem for quasi-log schemes do not directly produce the minimal modelprogram for quasi-log schemes or semi log canonical pairs. For somerelated examples, see [Ko11]. However, Kento Fujita ([Fk2]) estab-lished a variant of the minimal model program for semi-terminal pairsin order to construct semi-terminal modifications for quasi-projectivedemi-normal pairs. His arguments use not only the cone and contrac-tion theorem for semi log canonical pairs (see Theorem 6.7.4), but alsoKollar’s gluing theory (see [Ko13, Section 5]). For the details, see[Fk2].

6.8. On quasi-log Fano schemes

In this section, we discuss quasi-log Fano schemes and some relatedresults.

Let us introduce the notion of quasi-log Fano schemes.

Definition 6.8.1 (Quasi-log Fano schemes). Let [X,ω] be a quasi-log scheme and let π : X → S be a projective morphism betweenschemes. If −ω is π-ample, then [X,ω] is called a relative quasi-logFano scheme over S. When S is a point, we simply say that [X,ω] isa quasi-log Fano scheme.

The following result is an easy consequence of the adjunction andthe vanishing theorem for quasi-log schemes: Theorem 6.3.4.

Theorem 6.8.2 (see [Am1, Theorem 6.6]). Let [X,ω] be a quasi-log scheme and let π : X → S be a proper morphism between schemes

6.9. BASEPOINT-FREE THEOREM OF REID–FUKUDA TYPE 233

such that π∗OX ' OS and that −ω is nef and log big over S with respectto [X,ω]. Let P ∈ S be a closed point.

(i) Assume that X−∞ ∩ π−1(P ) 6= ∅ and C is a qlc stratum suchthat C ∩ π−1(P ) 6= ∅. Then C ∩X−∞ ∩ π−1(P ) 6= ∅.

(ii) Assume that [X,ω] has only qlc singularities, that is, X−∞ =∅. Then the set of all qlc strata intersecting π−1(P ) has aunique minimal element with respect to the inclusion.

Proof. Let C be a qlc stratum of [X,ω] such that P ∈ π(C) ∩π(X−∞). Then X ′ = C ∪ X−∞ with ω′ = ω|X′ is a quasi-log schemeand the restriction map π∗OX → π∗OX′ is surjective by Theorem 6.3.4.Since π∗OX ' OS, X−∞ and C intersect over a neighborhood of P . So,we have (i).

Assume that [X,ω] has only qlc singularities, that is, Nqlc(X,ω) =∅. Let C1 and C2 be two qlc strata of [X,ω] such that P ∈ π(C1)∩π(C2).The union X ′ = C1∪C2 with ω′ = ω|X′ is a qlc pair and the restrictionmap π∗OX → π∗OX′ is surjective. Therefore, C1 and C2 intersect overP . Furthermore, the intersection C1 ∩ C2 is a union of qlc strata byTheorem 6.3.7. Therefore, there exists a unique qlc stratum CP overa neighborhood of P such that CP ⊂ C for every qlc stratum C withP ∈ π(C). So, we finish the proof of (ii).

The following corollary is obvious by Theorem 6.8.2.

Corollary 6.8.3. Let (X,∆) be a proper log canonical pair. As-sume that −(KX +∆) is nef and log big with respect to (X,∆) and that(X,∆) is not klt. Then there exists a unique minimal log canonical cen-ter C0 such that every log canonical center contains C0. In particular,Nklt(X,∆) is connected.

For some related results, see [F36, Section 5]. In [F39], we obtain:

Theorem 6.8.4 (see [F39, Corollary 1.3]). Let [X,ω] be a quasi-logFano scheme with only qlc singularities. Then the algebraic fundamen-tal group of X is trivial, equivalently, X has nontrivial finite etalecovers.

We think that Theorem 6.8.4 is not so obvious. For the details ofTheorem 6.8.4 and some related results and conjectures, see [F39].

6.9. Basepoint-free theorem of Reid–Fukuda type

In this section, we explain the basepoint-free theorem of Reid–Fukuda type for quasi-log schemes. For the details, see [F40].

In [F40], we obtain:

234 6. FUNDAMENTAL THEOREMS FOR QUASI-LOG SCHEMES

Theorem 6.9.1 (Basepoint-free theorem of Reid–Fukuda type forquasi-log schemes). Let [X,ω] be a quasi-log scheme, let π : X → S bea projective morphism between schemes, and let L be a π-nef Cartierdivisor on X such that qL − ω is nef and log big over S with respectto [X,ω] for some positive real number q. Assume that OX−∞(mL) isπ-generated for every m 0. Then OX(mL) is π-generated for everym 0.

Remark 6.9.2. Theorem 6.9.1 was stated in [Am1, Theorem 7.2]without proof. Although Ambro wrote that the proof of [Am1, The-orem 7.2] is parallel to that of [Am1, Theorem 5.1], it does not seemto be true. For the details, see [F40, Remark 1.4].

By applying Theorem 6.9.1 to normal pairs, we obtain:

Theorem 6.9.3. Let X be a normal variety, let ∆ be an effectiveR-divisor on X such that KX +∆ is R-Cartier, and let π : X → S be aprojective morphism between schemes. Let L be a π-nef Cartier divisoron X such that qL− (KX +∆) is nef and log big over S with respect to(X,∆) for some positive real number q. Assume that ONlc(X,∆)(mL) isπ-generated for every m 0. Note that Nlc(X,∆) denotes the non-lclocus of (X,∆) and is defined by the non-lc ideal sheaf JNLC(X,∆) of(X,∆). Then OX(mL) is π-generated for every m 0.

As a special case, we have:

Corollary 6.9.4 (see [F17, Theorem 4.4]). Let (X,∆) be a logcanonical pair and let π : X → S be a projective morphism onto avariety S. Let L be a π-nef Cartier divisor on X such that qL− (KX +∆) is nef and log big over S with respect to (X,∆) for some positivereal number q. Then OX(mL) is π-generated for every m 0.

In this section, we prove Theorem 6.9.1 under the extra assumptionthat X−∞ = ∅, which is sufficient to Corollary 6.9.4.

First, let us prepare an easy lemma.

Lemma 6.9.5 (cf. [F40, Lemma 3.15]). Let [X,ω] be a quasi-logscheme with Nqlc(X,ω) = ∅ and let E be a finite R>0-linear combina-tion of effective Cartier divisors on X. We put

ω = ω + εE

with 0 < ε 1. Then [X, ω] has a natural quasi-log structure with thefollowing properties.

(i) Let Cii∈I be the set of qlc strata of [X,ω] contained in SuppE.We put

X♠ = ( ∪i∈ICi)

6.9. BASEPOINT-FREE THEOREM OF REID–FUKUDA TYPE 235

as in Theorem 6.3.4. Then Nqlc(X, ω) coincides with X♠

scheme theoretically.(ii) C is a qlc stratum of [X, ω] if and only if C is a qlc stratum

of [X,ω] with C 6⊂ SuppE.

Proof. Let f : (Y,BY ) → X be a quasi-log resolution as in Def-inition 6.2.2. By Proposition 6.3.1, the union of all strata of (Y,BY )mapped to X♠, which is denoted by Y ′′, is a union of some irreduciblecomponents of Y . We put Y ′ = Y − Y ′′ and

KY ′ +BY ′ = (KY +BY )|Y ′ .

We may further assume that (Y ′, BY ′ + f ∗E) is a globally embeddedsimple normal crossing pair by Proposition 6.3.1. We consider f :(Y ′, BY ′ + εf ∗E) → X with 0 < ε 1. We put A = d−(B<1

Y )e.Then X♠ is defined by the ideal sheaf f∗OY ′(A− Y ′′) (see the proof ofTheorem 6.3.4 (i)). Note that

(A− Y ′′)|Y ′ = −bBY ′ + εf ∗Ec+ (BY ′ + εf ∗E)=1

= d−(BY ′ + εf ∗E)<1e − b(BY ′ + εf ∗E)>1c.

Therefore, if we define Nqlc(X, ω) by the ideal sheaf

f∗OY ′(d−(BY ′ + εf ∗E)<1e − b(BY ′ + εf ∗E)>1c) = f∗OY ′(A− Y ′′),

then f : (Y ′, BY ′ + εf ∗E)→ X gives the desired quasi-log structure on[X, ω].

Let us prove Theorem 6.9.1 under the extra assumption thatX−∞ =∅.

Proof of Theorem 6.9.1 when X−∞ = ∅. We divide the proofinto several steps.

Step 1. If dimX = 0, then Theorem 6.9.1 obviously holds true.From now on, we assume that Theorem 6.9.1 holds for any quasi-logscheme Z with Z−∞ = ∅ and dimZ < dimX.

Step 2. We take a qlc stratum C of [X,ω]. We put X ′ = C. ThenX ′ has a natural quasi-log structure induced by [X,ω] (see Theorem6.3.4 (i)). By the vanishing theorem (see Theorem 6.3.4 (ii)), we haveR1π∗(IX′ ⊗OX(mL)) = 0 for every m ≥ q. Therefore, we obtain thatπ∗OX(mL) → π∗OX′(mL) is surjective for every m ≥ q. Thus, wemay assume that X is irreducible for the proof of Theorem 6.9.1 by

236 6. FUNDAMENTAL THEOREMS FOR QUASI-LOG SCHEMES

the following commutative diagram.

π∗π∗OX(mL) //

π∗π∗OX′(mL) //

0

OX(mL) // OX′(mL) // 0

Step 3. We may further assume that S is affine for the proof ofTheorem 6.9.1. Of course, we may assume that X is connected.

Step 4. In this step, we assume that X is the unique qlc stratumof [X,ω]. By Lemma 6.3.5, we have that X is normal. By Kodaira’slemma (see Lemma 2.1.18), we can write qL − ω ∼R A + E on Xsuch that A is a π-ample Q-divisor on X and E is a finite R>0-linearcombination of effective Cartier divisors on X. We put ω = ω + εEwith 0 < ε 1. Then [X, ω] is a quasi-log scheme with Nqlc(X, ω) = ∅(see Lemma 6.9.5). Note that

qL− ω ∼R (1− ε)(qL− ω) + εA

is π-ample. Therefore, by the basepoint-free theorem for quasi-logschemes (see Theorem 6.5.1), we obtain that OX(mL) is π-generatedfor every m 0.

Step 5. From now on, by Step 4, we may assume that there is aqlc center C ′ of [X,ω] (see Theorem 6.3.7 (i)). We put

X ′ = ( ∪i∈ICi)

as in Theorem 6.3.4, where Cii∈I is the set of all qlc centers of [X,ω],equivalently, X ′ = Nqklt(X,ω). Then, by induction on the dimen-sion, OX′(mL) is π-generated for every m 0. By the same argu-ments as in Step 2, that is, the surjectivity of the restriction mapπ∗OX(mL) → π∗OX′(mL) for every m ≥ q, OX(mL) is π-generatedin a neighborhood of X ′ for every large and positive integer m. Inparticular, for every prime number p and every large positive integer l,OX(plL) is π-generated in a neighborhood of X ′ = Nqklt(X,ω).

Step 6. In this step, we prove the following claim.

Claim. If the relative base locus Bsπ |plL| (with the reduced schemestructure) is not empty, then there is a positive integer a such thatBsπ |palL| is strictly smaller than Bsπ |plL|.

Proof of Claim. Note that Bsπ |palL| ⊆ Bsπ |plL| for every pos-itive integer a. Since qL− ω is nef and big over S, we can write

qL− ω ∼R A+ E

6.9. BASEPOINT-FREE THEOREM OF REID–FUKUDA TYPE 237

on X by Kodaira’s lemma (see, for example, Lemma 2.1.18, [F33,Lemma A.10], and so on) where A is a π-ample Q-divisor on X and Eis a finite R>0-linear combination of effective Cartier divisors on X. Wenote that X is projective over S and that X is not necessarily normal.By Lemma 6.9.5, we have a new quasi-log structure on [X, ω], whereω = ω + εE with 0 < ε 1, such that

Nqlc(X, ω) = ( ∪i∈ICi)

where Cii∈I is the set of qlc centers of [X,ω] contained in SuppE.We put n = dimX. Let D1, · · · , Dn+1 be general members of |plL|.

Note that OX(plL) is π-generated in a neighborhood of Nqklt(X,ω).Let f : (Y,BY ) → X be a quasi-log resolution of [X, ω]. We considerf : (Y,BY +

∑n+1i=1 f

∗Di) → X. We may assume that (Y, SuppBY +∑n+1i=1 f

∗Di) is a globally embedded simple normal crossing pair byProposition 6.3.1. By Step 5, we can take the minimal positive realnumber c such that BY + c

∑n+1i=1 f

∗Di is a subboundary R-divisor overX \ Nqlc(X, ω). Note that we have c < 1 by Lemma 6.3.9. Thus,

f : (Y,BY + cn+1∑i=1

f ∗Di)→ X

gives a natural quasi-log structure on [X, ω + c∑n+1

i=1 Di]. Note that

[X, ω + c∑n+1

i=1 Di] has only quasi-log canonical singularities on X \Nqlc(X, ω). We also note that Di is a general member of |plL| forevery i. By construction, there is a qlc center C0 of [X, ω+ c

∑n+1i=1 Di]

contained in Bsπ |plL|. We put ω + c∑n+1

i=1 Di = ω. Then

C0 ∩ Nqlc(X,ω) = ∅because

Bsπ |plL| ∩ Nqklt(X,ω) = ∅.Note that Nqlc(X,ω) = Nqlc(X, ω) by construction. We also note that

(q + cpl)L− ω ∼R (1− ε)(qL− ω) + εA

is ample over S. Therefore,

π∗OX(mL)→ π∗OC0(mL)⊕ π∗ONqlc(X,ω)(mL)

is surjective for every m ≥ q + cpl since

R1π∗(IC0∪Nqklt(X,ω) ⊗OX(mL)) = 0

for every m ≥ q + cpl by Theorem 6.3.4 (ii). Moreover, OC0(mL) isπ-generated for every m 0 by the basepoint-free theorem for quasi-log schemes (see Theorem 6.5.1). Note that [C0, ω|C0 ] is a quasi-log

238 6. FUNDAMENTAL THEOREMS FOR QUASI-LOG SCHEMES

scheme with only quasi-log canonical singularities by Theorem 6.3.4 (i)and Lemma 6.3.8. Therefore, we can construct a section s of OX(palL)for some positive integer a such that s|C0 is not zero and s is zeroon Nqlc(X,ω). Thus, Bsπ |palL| is strictly smaller than Bsπ |plL|. Wecomplete the proof of Claim.

Step 7. By Step 6 and the noetherian induction, OX(plL) andOX(p′l

′L) are both π-generated for large l and l′, where p and p′ are

distinct prime numbers. So, there exists a positive integer m0 such thatOX(mL) is π-generated for every m ≥ m0.

Thus we obtain the desired basepoint-free theorem. For the proof of Theorem 6.9.1 with X−∞ 6= ∅, we need various new

operations on quasi-log schemes. The proof of Theorem 6.9.1 in [F40]is much harder than the proof given in this section. For the details, see[F40].

CHAPTER 7

Some supplementary topics

In this chapter, we treat some related results and supplementarytopics.

In Section 7.1, we discuss Alexeev’s criterion for Serre’s S3 condition(see [Ale3]) with slight generalizations. Note that Alexeev’s criterion isa clever application of our new torsion-free theorem (see Theorem 5.6.3(i) or Theorem 3.16.3 (i)). Although we have already obtained vari-ous related results and several generalizations (see, for example, [AH],[Ko12], [Kv6], and so on), we only treat Alexeev’s criterion here. Notethat log canonical singularities are not necessarily Cohen–Macaulay. InSection 7.2, we collect some basic properties of cone singularities forthe reader’s convenience. The results in Section 7.2 are useful when weconstruct various examples. We have already used them several timesin this book. In Section 7.3, we give some examples of threefolds. Theyshow that we need flips even when we run the minimal model programfor a smooth projective threefold with the unique smooth projectiveminimal model. This means that we necessarily have singular varietiesin the intermediate step of the above minimal model program. In Sec-tion 7.4, we describe an explicit example of threefold toric log flip. Itmay help us understand the proof of the special termination theorem in[F13]. In Section 7.5, we explicitly construct a three-dimensional non-Q-factorial canonical Gorenstein toric flip. It may help us understandthe non-Q-factorial minimal model program explained in Section 4.9.In this example, the flipped variety is smooth and the Picard numberincreases by a flip.

7.1. Alexeev’s criterion for S3 condition

In this section, we explain Alexeev’s criterion for Serre’s S3 condi-tion (see Theorem 7.1.1). It is a clever application of Theorem 5.6.3(i) (see also Theorem 3.16.3 (i)). In general, log canonical singularitiesare not Cohen–Macaulay. So, the results in this section will be usefulfor the study of log canonical pairs.

Theorem 7.1.1 (cf. [Ale3, Lemma 3.2]). Let (X,B) be a log canon-ical pair with dimX = n ≥ 3 and let P ∈ X be a scheme theoretic point

239

240 7. SOME SUPPLEMENTARY TOPICS

such that dim P ≤ n − 3. Assume that P is not a log canonicalcenter of (X,B). Then the local ring OX,P satisfies Serre’s S3 condi-tion.

We slightly changed the original formulation. The following proofis essentially the same as Alexeev’s. We use local cohomologies tocalculate depths.

Proof. We note that OX,P satisfies Serre’s S2 condition because Xis normal. Since the assertion is local, we may assume that X is affine.Let f : Y → X be a resolution of X such that Exc(f) ∪ Supp f−1

∗ B isa simple normal crossing divisor on Y . Then we can write

KY +BY = f∗(KX +B)

such that SuppBY is a simple normal crossing divisor on Y . We putA = d−(B<1

Y )e ≥ 0. Then we obtain

A = KY +B=1Y + BY − f∗(KX +B).

Therefore, by Theorem 5.6.3 (i) or Theorem 3.16.3 (i), every associ-ated prime of R1f∗OY (A) is the generic point of some log canonicalcenter of (X,B). Thus, P is not an associated prime of R1f∗OY (A) byassumption.

We put XP = SpecOX,P and YP = Y ×X XP . Then P is a closedpoint of XP and it is sufficient to prove that H2

P (XP ,OXP) = 0. We

put F = f−1(P ), where f : YP → XP . Then we have the followingvanishing theorem. It is nothing but Lemma 3.15.2 when P is a closedpoint of X (see [Har4, Chapter III, Exercise 2.5]).

Lemma 7.1.2 (cf. Lemma 3.15.2). We have H iF (YP ,OYP

) = 0 for

i < n− dim P.Proof of Lemma 7.1.2. Let I denote an injective hull ofOXP

/mP

as an OXP-module, where mP is the maximal ideal corresponding to

P . We have

RΓFOYP' RΓP (Rf∗OYP

)

' Hom(RHom(Rf∗OYP, ω•XP

), I)

' Hom(Rf∗OY (KY )⊗OXP[n−m], I),

where m = dim P, by the local duality theorem ([Har1, Chapter V,Theorem 6.2]) and Grothendieck duality ([Har1, Chapter VII, The-orem 3.3]). We note the shift that normalizes the dualizing complexω•XP

. Therefore, we obtain H iF (YP ,OYP

) = 0 for i < n − m because

Rjf∗OY (KY ) = 0 for every j > 0 by the Grauert–Riemenschneidervanishing theorem (see Theorem 3.2.7).

7.1. ALEXEEV’S CRITERION FOR S3 CONDITION 241

Let us go back to the proof of the theorem. We use the methodof two spectral sequences discussed in Section 3.15. We consider thefollowing spectral sequences

Ep,q2 = Hp

P (XP , Rqf∗OYP

(A))⇒ Hp+qF (YP ,OYP

(A)),

and′Ep,q

2 = HpP (XP , R

qf∗OYP)⇒ Hp+q

F (YP ,OYP).

By the above spectral sequences, we have the commutative diagram.

H2F (YP ,OYP

) // H2F (YP ,OYP

(A))

H2P (XP , f∗OYP

)

OO

H2P (XP , f∗OYP

(A))

φ

OO

H2P

(XP ,OXP

) H2P

(XP ,OXP

)

Since P is not an associated prime of R1f∗OY (A), we have

E0,12 = H0

P (XP , R1f∗OYP

(A)) = 0.

By the edge sequence

0→ E1,02 → E1 → E0,1

2 → E2,02

φ→ E2 → · · · ,

we know that φ : E2,02 → E2 is injective. Therefore, H2

P (XP ,OXP) →

H2F (YP ,OYP

) is injective by the above big commutative diagram. Thus,we obtain H2

P (XP ,OXP) = 0 since H2

F (YP ,OYP) = 0 by Lemma 7.1.2.

Remark 7.1.3. The original argument in the proof of [Ale3, Lemma3.2] has some compactification problems when X is not projective. Ourproof does not need any compactifications of X.

As an easy application of Theorem 7.1.1, we have the followingresult. It is [Ale3, Theorem 3.4].

Theorem 7.1.4 (see [Ale3, Theorem 3.4]). Let (X,B) be a logcanonical pair and let D be an effective Cartier divisor. Assume thatthe pair (X,B + εD) is log canonical for some ε > 0. Then D is S2.

Proof. Without loss of generality, we may assume that dimX =n ≥ 3. Let P ∈ D ⊂ X be a scheme theoretic point such thatdim P ≤ n−3. We localize X at P and assume that X = SpecOX,P .

242 7. SOME SUPPLEMENTARY TOPICS

By assumption, P is not a log canonical center of (X,B). By The-orem 7.1.1, we obtain that H i

P (X,OX) = 0 for i < 3. Therefore,H iP (D,OD) = 0 for i < 2 by the long exact sequence

· · · → H iP (X,OX(−D))→ H i

P (X,OX)→ H iP (D,OD)→ · · · .

We note that H iP (X,OX(−D)) ' H i

P (X,OX) = 0 for i < 3. Thus, Dsatisfies Serre’s S2 condition.

We give a supplement to adjunction (see Theorem 6.3.4 (i)). It maybe useful for the study of limits of stable pairs (see [Ale3]).

Theorem 7.1.5 (Adjunction for Cartier divisors on log canonicalpairs). Let (X,B) be a log canonical pair and let D be an effectiveCartier divisor on X such that (X,B+D) is log canonical. Let V be aunion of log canonical centers of (X,B). We consider V as a reducedclosed subscheme of X. We define a scheme structure on V ∩D by thefollowing short exact sequence

0→ OV (−D)→ OV → OV ∩D → 0.

Then, OV ∩D is reduced and semi-normal.

Proof. First, we note that V ∩D is a union of log canonical centersof (X,B + D) (see Theorem 6.3.7). We also note that D contains nolog canonical centers of (X,B) since (X,B +D) is log canonical. Letf : Y → X be a resolution such that Exc(f) ∪ Supp f−1

∗ (B + D) is asimple normal crossing divisor on Y . We can write

KY +BY = f ∗(KX +B +D)

such that SuppBY is a simple normal crossing divisor on Y . We takemore blow-ups and may assume that f−1(V ∩D) and f−1(V ) are simplenormal crossing divisors. Then the union of all strata of B=1

Y mappedto V ∩D (resp. V ), which is denoted by W (resp. U+W ), is a divisor onY . We put A = d−(B<1

Y )e ≥ 0 and consider the following commutativediagram.

0 // OY (A− U −W )

// OY (A) // OU+W (A)

// 0

0 // OY (A−W ) // OY (A) // OW (A) // 0

7.1. ALEXEEV’S CRITERION FOR S3 CONDITION 243

By applying f∗, we obtain the big commutative diagram by Theorem5.6.3 (i) and Theorem 6.3.4 (i).

0

0

f∗OU(A−W )

0 // f∗OY (A− U −W )

// OX // OV

// 0

0 // f∗OY (A−W )

// OX // OV ∩D

// 0

f∗OU(A−W )

0

0

A key point is that the connecting homomorphism

f∗OU(A−W )→ R1f∗OY (A− U −W )

is a zero map by Theorem 5.6.3 (i). We note that OV and OV ∩D in theabove diagram are the structure sheaves of qlc pairs [V, (KX+B+D)|V ]and [V ∩D, (KX +B +D)|V ∩D] induced by (X,B +D) (see Theorem6.3.4 (i)). In particular, OV ' f∗OU+W and OV ∩D ' f∗OW . So, OVand OV ∩D are reduced and semi-normal since W and U+W are simplenormal crossing divisors on Y .

Therefore, to prove this theorem, it is sufficient to see that f∗OU(A−W ) ' OV (−D). We can write

A = KY +B=1Y + BY − f ∗(KX +B +D)

andf ∗D = W + E + f−1

∗ D,

where E is an effective f -exceptional divisor. We note that f−1∗ D∩U =

∅. Since A − W = A − f ∗D + E + f−1∗ D, it is enough to see that

f∗OU(A+E+f−1∗ D) ' f∗OU(A+E) ' OV . We consider the following

short exact sequence

0→ OY (A+ E − U)→ OY (A+ E)→ OU(A+ E)→ 0.

Note that

A+ E − U = KY +B=1Y − f−1

∗ D − U −W + BY − f∗(KX +B).

244 7. SOME SUPPLEMENTARY TOPICS

Thus, the connecting homomorphism f∗OU(A + E) → R1f∗OY (A +E − U) is a zero map by Theorem 5.6.3 (i). Therefore, we obtain that

0→ f∗OY (A+ E − U)→ OX → f∗OU(A+ E)→ 0.

We can easily check that f∗OY (A + E − U) = IV , the defining idealsheaf of V . So, we have f∗OU(A + E) ' OV . We finish the proof ofthis theorem.

The next corollary is one of the main results in [Ale3]. The originalproof in [Ale3] depends on the S2-fication. Our proof uses adjunction(see Theorem 7.1.5). As a consequence, we obtain the semi-normalityof bBc ∩D.

Corollary 7.1.6 (cf. [Ale3, Theorem 4.1]). Let (X,B) be a logcanonical pair and let D be an effective Cartier divisor on D such that(X,B +D) is log canonical. Then D is S2 and the scheme bBc ∩D isreduced and semi-normal.

Proof. By Theorem 7.1.4, D satisfies Serre’s S2 condition. ByTheorem 7.1.5, bBc ∩D is reduced and semi-normal.

The following proposition may be useful. So, we include it here. Itis [Ale3, Lemma 3.1] with slight modifications as Theorem 7.1.1.

Proposition 7.1.7 (cf. [Ale3, Lemma 3.1]). Let X be a normalvariety with dimX = n ≥ 3 and let f : Y → X be a resolutionof singularities. Let P ∈ X be a scheme theoretic point such thatdim P ≤ n − 3. Then the local ring OX,P is S3 if and only if P isnot an associated prime of R1f∗OY .

Proof. We put XP = SpecOX,P , YP = Y ×X XP , and F =f−1(P ), where f : YP → XP . We consider the following spectralsequence

Ei,j2 = H i

P (X,Rjf∗OYP)⇒ H i+j

F (YP ,OYP).

Since H1F (YP ,OYP

) = H2F (YP ,OYP

) = 0 by Lemma 7.1.2, we have anisomorphismH0

P (XP , R1f∗OYP

) ' H2P (XP ,OXP

). Therefore, the depthof OX,P is ≥ 3 if and only if H2

P (XP ,OXP) = H0

P (XP , R1f∗OYP

) = 0.It is equivalent to the condition that P is not an associated prime ofR1f∗OY .

7.1.8 (Supplements). Here, we give a slight generalization of [Ale3,Theorem 3.5]. We can prove it by a similar method to the proof ofTheorem 7.1.1.

Theorem 7.1.9 (cf. [Ale3, Theorem 3.5]). Let (X,B) be a logcanonical pair and let D be an effective Cartier divisor on X such

7.1. ALEXEEV’S CRITERION FOR S3 CONDITION 245

that (X,B + εD) is log canonical for some ε > 0. Let V be a unionof some log canonical centers of (X,B). We consider V as a reducedclosed subscheme of X. We can define a scheme structure on V ∩ Dby the following exact sequence

0→ OV (−D)→ OV → OV ∩D → 0.

Then the scheme V ∩ D satisfies Serre’s S1 condition. In particular,bBc ∩D has no embedded point.

Proof. Without loss of generality, we may assume thatX is affine.We take a resolution f : Y → X such that Exc(f) ∪ Supp f−1

∗ B is asimple normal crossing divisor on Y . Then we can write

KY +BY = f∗(KX +B)

such that SuppBY is a simple normal crossing divisor on Y . We takemore blow-ups and may assume that the union of all strata of B=1

Y

mapped to V , which is denoted by W , is a divisor on Y . Moreover, forany log canonical center C of (X,B) contained in V , we may assumethat f−1(C) is a divisor on Y . We consider the following short exactsequence

0→ OY (A−W )→ OY (A)→ OW (A)→ 0,

where A = d−(B<1Y )e ≥ 0. By taking higher direct images, we obtain

0→ f∗OY (A−W )→ OX → f∗OW (A)→ R1f∗OY (A−W )→ · · · .

By Theorem 5.6.3 (i) or Theorem 3.16.3 (i), we have that f∗OW (A)→R1f∗OY (A−W ) is a zero map, f∗OW (A) ' OV , and f∗OY (A−W ) 'IV , the defining ideal sheaf of V on X. We note that f∗OW ' OV .In particular, OV is reduced and semi-normal. For the details, seeTheorem 6.3.4 (i).

Let P ∈ V ∩ D be a scheme theoretic point such that the heightof P in OV ∩D is ≥ 1. We may assume that dimV ≥ 2 around P .Otherwise, the theorem is trivial. We put VP = SpecOV,P , WP =W ×V VP , and F = f−1(P ), where f : WP → VP . The pull-back ofD on VP is denoted by D for simplicity. To check this theorem, itis sufficient to see that H0

P (VP ∩ D,OVP∩D) = 0. First, we note thatH0P (VP ,OVP

) = H0F (WP ,OWP

) = 0 by f∗OW ' OV . Next, as in theproof of Lemma 7.1.2, we have

RΓFOWP' RΓP (Rf∗OWP

)

' Hom(RHom(Rf∗OWP, ω•VP

), I)

' Hom(Rf∗OW (KW )⊗OVP[n− 1−m], I),

246 7. SOME SUPPLEMENTARY TOPICS

where n = dimX, m = dim P, and I is an injective hull of OVP/mP

as an OVP-module such that mP is the maximal ideal corresponding to

P . Once we obtainRn−m−2f∗OW (KW )⊗OVP= 0, thenH1

F (WP ,OWP) =

0. It implies thatH1P (VP ,OVP

) = 0 sinceH1P (VP ,OVP

) ⊂ H1F (WP ,OWP

).By the long exact sequence

· · · → H0P (VP ,OVP

)→ H0P (VP ∩D,OVP∩D)

→ H1P (VP ,OVP

(−D))→ · · · ,

we obtain H0P (VP ∩D,OVP∩D) = 0. This is because H0

P (VP ,OVP) = 0

and H1P (VP ,OVP

(−D)) ' H1P (VP ,OVP

) = 0. So, it is sufficient to seethat Rn−m−2f∗OW (KW )⊗OVP

= 0.To check the vanishing of Rn−m−2f∗OW (KW )⊗OVP

, by taking gen-eral hyperplane cuts m times, we may assume that m = 0 and P ∈ Xis a closed point. We note that the dimension of any irreducible com-ponent of V passing through P is ≥ 2 since P is not a log canonicalcenter of (X,B) (see Theorem 6.3.7).

On the other hand, we can write W = U1 + U2 such that U2 isthe union of all the irreducible components of W whose images by fhave dimensions ≥ 2 and U1 = W − U2. We note that the dimensionof the image of any stratum of U2 by f is ≥ 2 by the construction off : Y → X. We consider the following exact sequence

· · · → Rn−2f∗OU2(KU2)→ Rn−2f∗OW (KW )

→ Rn−2f∗OU1(KU1 + U2|U1)→ Rn−1f∗OU2(KU2)→ · · · .

We have Rn−2f∗OU2(KU2) = Rn−1f∗OU2(KU2) = 0 around P since thedimension of general fibers of f : U2 → f(U2) is ≤ n − 3. Thus, weobtain Rn−2f∗OW (KW ) ' Rn−2f∗OU1(KU1 + U2|U1) around P . There-fore, the support of Rn−2f∗OW (KW ) around P is contained in one-dimensional log canonical centers of (X,B) in V and Rn−2f∗OW (KW )has no zero-dimensional associated prime around P by Theorem 5.6.3(i). Note that, by the above argument, we have Rn−2f∗OW (KW ) = 0when U1 = 0. When U1 6= 0, by taking a general hyperplane cut of Xagain, we have the vanishing of Rn−2f∗OW (KW ) around P by Lemma7.1.10 below. So, we finish the proof.

We have already used the following lemma in the proof of Theorem7.1.9.

Lemma 7.1.10. Let (Z,∆) be a d-dimensional log canonical pairand let x ∈ Z be a closed point such that x is a log canonical center of(Z,∆). Let V be a union of some log canonical centers of (Z,∆) suchthat dimV > 0, x ∈ V , and x is not isolated in V . Let f : Y → Z be

7.2. CONE SINGULARITIES 247

a resolution such that f−1(x) and f−1(V ) are divisors on Y and thatExc(f)∪Supp f−1

∗ ∆ is a simple normal crossing divisor on Y . We canwrite

KY +BY = f ∗(KZ + ∆)

such that SuppBY is a simple normal crossing divisor on Y . Let W bethe union of all the irreducible components of B=1

Y mapped to V . ThenRd−1f∗OW (KW ) = 0 around x.

Proof. We can write W = W1 +W2, where W2 is the union of allthe irreducible components of W mapped to x by f and W1 = W −W2.We consider the following short exact sequence

0→ OY (KY )→ OY (KY +W )→ OW (KW )→ 0.

By the Grauert–Riemenschneider vanishing theorem (see Theorem 3.2.7),we obtain that

Rd−1f∗OY (KY +W ) ' Rd−1f∗OW (KW ).

Next, we consider the short exact sequence

0→ OY (KY +W1)→ OY (KY +W )→ OW2(KW2 +W1|W2)→ 0.

Around x, the image of any irreducible component ofW1 by f is positivedimensional. Therefore, Rd−1f∗OY (KY + W1) = 0 near x. It can bechecked by induction on the number of irreducible components usingthe following exact sequence

· · · → Rd−1f∗OY (KY +W1 − S)→ Rd−1f∗OY (KY +W1)

→ Rd−1f∗OS(KS + (W1 − S)|S)→ · · · ,where S is an irreducible component of W1. On the other hand, wehave

Rd−1f∗OW2(KW2 +W1|W2) ' Hd−1(W2,OW2(KW2 +W1|W2))

and Hd−1(W2,OW2(KW2 + W1|W2)) is dual to H0(W2,OW2(−W1|W2)).Note that f∗OW2 ' Ox and f∗OW ' OV by the usual argument onadjunction (see Theorem 6.3.4 (i)). Since W2 and W = W1 + W2 areconnected over x, H0(W2,OW2(−W1|W2)) = 0. We note thatW1|W2 6= 0since x is not isolated in V . This means that Rd−1f∗OW (KW ) = 0around x by the above arguments.

7.2. Cone singularities

In this section, we collect some basic facts on cone singularities forthe reader’s convenience. They are useful when we construct examples.

First, let us give two useful lemmas.

248 7. SOME SUPPLEMENTARY TOPICS

Lemma 7.2.1. Let X be an n-dimensional normal variety and letf : Y → X be a resolution of singularities. Assume that Rif∗OY = 0for 1 ≤ i ≤ n− 2. Then X is Cohen–Macaulay.

Proof. We may assume that n ≥ 3. Since SuppRn−1f∗OY is zero-dimensional, we may assume that there exists a closed point x ∈ Xsuch that X has only rational singularities outside x by shrinking Xaround x. Therefore, it is sufficient to see that the depth of OX,x is≥ n = dimX. We consider the following spectral sequence

Ei,j2 = H i

x(X,Rjf∗OY )⇒ H i+j

F (Y,OY ),

where F = f−1(x). Then H ix(X,OX) = Ei,0

2 ' Ei,0∞ = 0 for i ≤ n − 1.

This is because H iF (Y,OY ) = 0 for i ≤ n − 1 by Lemma 3.15.2. This

means that the depth of OX,x is ≥ n. So, we obtain that X is Cohen–Macaulay.

Lemma 7.2.2. Let X be an n-dimensional normal variety and letf : Y → X be a resolution of singularities. Let x ∈ X be a closedpoint. Assume that X is Cohen–Macaulay and that X has only rationalsingularities outside x. Then Rif∗OY = 0 for 1 ≤ i ≤ n− 2.

Proof. We may assume that n ≥ 3. By assumption, we haveSuppRif∗OY ⊂ x for 1 ≤ i ≤ n − 1. We consider the followingspectral sequence

Ei,j2 = H i

x(X,Rjf∗OY )⇒ H i+j

F (Y,OY ),

where F = f−1(x). Then H0x(X,R

jf∗OY ) = E0,j2 ' E0,j

∞ = 0 forj ≤ n − 2 since Ei,j

2 = 0 for i > 0 and j > 0, Ei,02 = 0 for i ≤ n − 1,

and HjF (Y,OY ) = 0 for j < n by Lemma 3.15.2. Therefore, we obtain

Rif∗OY = 0 for 1 ≤ i ≤ n− 2. We point out the following fact explicitly for the reader’s conve-

nience. It is [Ko8, 11.2 Theorem. (11.2.5)].

Lemma 7.2.3. Let f : Y → X be a proper morphism, let x ∈ Xbe a closed point, and let G be a sheaf on Y . If SuppRjf∗G ⊂ xfor 1 ≤ i < k and H i

F (Y,G) = 0 for i ≤ k where F = f−1(x), thenRjf∗G ' Hj+1

x (X, f∗G) for j = 1, · · · , k − 1.

The assumptions in Lemma 7.2.2 are satisfied for n-dimensional iso-lated Cohen–Macaulay singularities. Therefore, we have the followingcorollary of Lemmas 7.2.1 and 7.2.2.

Corollary 7.2.4. Let x ∈ X be an n-dimensional normal isolatedsingularity. Then x ∈ X is Cohen–Macaulay if and only if Rif∗OY = 0for 1 ≤ i ≤ n− 2, where f : Y → X is a resolution of singularities.

7.2. CONE SINGULARITIES 249

We note the following easy example.

Example 7.2.5. Let V be a cone over a smooth plane cubic curveand let ϕ : W → V be the blow-up at the vertex. Then W is smoothand KW = ϕ∗KV − E, where E is an elliptic curve. In particular,V is log canonical. Let C be a smooth curve. We put Y = W × C,X = V × C, and f = ϕ × idC : Y → X, where idC is the identitymap of C. By construction, X is a log canonical threefold. We caneasily check that X is Cohen–Macaulay (see also Theorem 7.1.1 andProposition 7.1.7). We note that R1f∗OY 6= 0 and that R1f∗OY hasno zero-dimensional associated components. Therefore, the Cohen–Macaulayness ofX does not necessarily imply the vanishing of R1f∗OY .

Let us go to cone singularities (see also [Ko8, 3.8 Example] and[Ko10, Exercises 70, 71]).

Lemma 7.2.6 (Projective normality). Let X ⊂ PN be a normalprojective irreducible variety and let V ⊂ AN+1 be the cone over X.Then V is normal if and only if

H0(PN ,OPN (m))→ H0(X,OX(m))

is surjective for every m ≥ 0. In this case, X ⊂ PN is said to beprojectively normal.

Proof. Without loss of generality, we may assume that dimX ≥ 1.Let P ∈ V be the vertex of V . By construction, we have H0

P (V,OV ) =0. We consider the following commutative diagram.

0 // H0(AN+1,OAN+1)

// H0(AN+1 \ P,OAN+1)

// 0

0 // H0(V,OV )

// H0(V \ P,OV ) // H1P (V,OV ) // 0

0

We note that H i(V,OV ) = 0 for every i > 0 since V is affine. Bythe above commutative diagram, it is easy to see that the followingconditions are equivalent.

(a) V is normal.(b) the depth of OV,P is ≥ 2.(c) H1

P (V,OV ) = 0.(d) H0(AN+1 \ P,OAN+1)→ H0(V \ P,OV ) is surjective.

250 7. SOME SUPPLEMENTARY TOPICS

The condition (d) is equivalent to the condition thatH0(PN ,OPN (m))→H0(X,OX(m)) is surjective for every m ≥ 0. We note that

H0(AN+1 \ P,OAN+1) '⊕m≥0

H0(PN ,OPN (m))

andH0(V \ P,OV ) '

⊕m≥0

H0(X,OX(m)).

So, we finish the proof. The next lemma is more or less well known to the experts.

Lemma 7.2.7. Let X ⊂ PN be a normal projective irreducible vari-ety and let V ⊂ AN+1 be the cone over X. Assume that X is projec-tively normal and that X has only rational singularities. Then we havethe following properties.

(1) V is Cohen–Macaulay if and only if H i(X,OX(m)) = 0 forevery 0 < i < dimX and m ≥ 0.

(2) V has only rational singularities if and only if H i(X,OX(m)) =0 for every i > 0 and m ≥ 0.

Proof. We put n = dimX and may assume n ≥ 1. For (1), itis sufficient to prove that H i

P (V,OV ) = 0 for 2 ≤ i ≤ n if and only ifH i(X,OX(m)) = 0 for every 0 < i < n and m ≥ 0 since V is normal,where P ∈ V is the vertex of V . Let f : W → V be the blow-upat P and E ' X the exceptional divisor of f . We note that W isthe total space of OX(−1) over E ' X and that W has only rationalsingularities. Since V is affine, we obtain H i(V \P,OV ) ' H i+1

P (V,OV )for every i ≥ 1. Since W has only rational singularities, we haveH iE(W,OW ) = 0 for i < n+ 1 (see Lemma 3.15.2 and Remark 3.15.3).

Therefore,

H i(V \ P,OV ) ' H i(W \ E,OW ) ' H i(W,OW )

for i ≤ n− 1. Thus,

H iP (V,OV ) ' H i−1(V \P,OV ) ' H i−1(W,OW ) '

⊕m≥0

H i−1(X,OX(m))

for 2 ≤ i ≤ n. So, we obtain the desired equivalence.For (2), we consider the following commutative diagram.

0 // Hn(V \ P,OV )

'

// Hn+1P (V,OV )

α

// 0

0 // Hn(W,OW ) // Hn(W \ E,OW ) // Hn+1E (W,OW )

7.3. FRANCIA’S FLIP REVISITED 251

We note that V is Cohen–Macaulay if and only if Rif∗OW = 0 for1 ≤ i ≤ n− 1 (see Lemmas 7.2.1 and 7.2.2) since W has only rationalsingularities. From now on, we assume that V is Cohen–Macaulay.Then, V has only rational singularities if and only if Rnf∗OW = 0. Bythe same argument as in the proof of Theorem 3.15.1, the kernel of αis H0

P (V,Rnf∗OW ). Thus, Rnf∗OW = 0 if and only if Hn(W,OW ) '⊕m≥0H

n(X,OX(m)) = 0 by the above commutative diagram. So, weobtain the statement (2) with the aid of (1).

The following proposition is very useful when we construct variousexamples. We have already used it several times in this book.

Proposition 7.2.8. Let X ⊂ PN be a normal projective irreduciblevariety and let V ⊂ AN+1 be the cone over X. Assume that X isprojectively normal. Let ∆ be an effective R-divisor on X and let B bethe cone over ∆. Then, we have the following properties.

(1) KV + B is R-Cartier if and only if KX + ∆ ∼R rH for somer ∈ R, where H ⊂ X is the hyperplane divisor on X ⊂ PN .

(2) If KX + ∆ ∼R rH, then (V,B) is(a) terminal if and only if r < −1 and (X,∆) is terminal,(b) canonical if and only if r ≤ −1 and (X,∆) is canonical,(c) klt if and only if r < 0 and (X,∆) is klt, and(d) lc if and only if r ≤ 0 and (X,∆) is lc.

Proof. Let f : W → V be the blow-up at the vertex P ∈ V andE ' X the exceptional divisor of f . If KV + B is R-Cartier, thenKW + f−1

∗ B ∼R f ∗(KV + B) + aE for some a ∈ R. By restricting itto E, we obtain that KX + ∆ ∼R −(a + 1)H. On the other hand, ifKX +∆ ∼R rH, then KW + f−1

∗ B ∼R,f −(r+1)E. Therefore, KV +Bis R-Cartier on V . Thus, we have the statement (1). For (2), it issufficient to note that

KW + f−1∗ B = f ∗(KV +B)− (r + 1)E

and that W is the total space of OX(−1) over E ' X.

7.3. Francia’s flip revisited

In this section, we treat some explicit examples of threefolds. Theseexamples show that the notion of flips is indispensable for the study ofhigher-dimensional algebraic varieties.

In Example 7.3.1, we construct Francia’s flip on a projective toricvariety explicitly. Francia’s flip is a monumental example (see [Fra]).So, we include it here. Our description may look slightly different fromthe usual one because we use the toric geometry.

252 7. SOME SUPPLEMENTARY TOPICS

Example 7.3.1. We fix a lattice N ' Z3 and consider the latticepoints e1 = (1, 0, 0), e2 = (0, 1, 0), e3 = (0, 0, 1), e4 = (1, 1,−2), ande5 = (−1,−1, 1). First, we consider the complete fan ∆1 spannedby e1, e2, e4, and e5. Since e1 + e2 + e4 + 2e5 = 0, X1 = X(∆1) isP(1, 1, 1, 2). Next, we take the blow-up f : X2 = X(∆2) → X1 alongthe ray e3 = (0, 0, 1). Then X2 is a projective Q-factorial toric varietywith only one 1

2(1, 1, 1)-singular point. Since ρ(X2) = 2, we have one

more contraction morphism ϕ : X2 → X3 = X(∆3). This contractionmorphism ϕ corresponds to the removal of the wall 〈e1, e2〉 from ∆2.We can easily check that ϕ is a flipping contraction. By adding thewall 〈e3, e4〉 to ∆3, we obtain the following flipping diagram.

X2//_______

ϕ !!BBB

BBBB

BX4

ψ||||

||||

X3

It is an example of Francia’s flip. Note that e3 + e4 + e5 = 0 ande1 + e2 = 2e3 + 1e4 + 0e5. Therefore, we can easily check that X4 'PP1(OP1⊕OP1(1)⊕OP1(2)) (see, for example, [Ful, Exercise in Section2.4]). We can also check that the flipped curve C ' P1 is the sectionof π : PP1(OP1 ⊕ OP1(1) ⊕ OP1(2)) → P1 defined by the projectionOP1 ⊕OP1(1)⊕OP1(2)→ OP1 → 0.

By taking double covers, we have an interesting example of smoothprojective threefolds (cf. [Fra]).

Example 7.3.2. We use the same notation as in Example 7.3.1. Letg : X5 → X2 be the blow-up along the ray e6 = (1, 1,−1). Then X5 isa smooth projective toric variety. Let OX4(1) be the tautological linebundle of the P2-bundle π : X4 = PP1(OP1 ⊕ OP1(1) ⊕ OP1(2)) → P1.It is easy to see that OX4(1) is nef and OX4(1) · C = 0, where Cis the flipped curve. Therefore, there exists a line bundle L on X3

such that OX4(1) ' ψ∗L, where ψ : X4 → X3. We take a generalmember D ∈ |L⊗8|. We note that |L| is free since L is nef. We takea double cover X → X4 (resp. Y → X5) ramifying along Suppψ−1D(resp. Supp(ϕ g)−1D). Then X is a smooth projective variety suchthat KX is ample. It is obvious that Y is a smooth projective varietyand is birational to X. So, X is a smooth projective threefold withample canonical divisor and is the unique minimal model of a smoothprojective threefold Y . It is easy to see that we need flips to obtainthe minimal model X from Y by running a minimal model program.In particular, the minimal model program from Y to X must pass

7.4. A SAMPLE COMPUTATION OF A LOG FLIP 253

through singular varieties by Theorem 1.1.4. Note that Y 99K X is nota morphism.

Example 7.3.2 clarifies the difference between the minimal modeltheory for smooth projective surfaces and the minimal model programfor higher-dimensional algebraic varieties.

7.4. A sample computation of a log flip

In this section, we treat an example of threefold toric log flips. Ingeneral, it is difficult to know what happens around the flipping curve.Therefore, the following nontrivial example is valuable because we cansee the behavior of the flip explicitly. It helps us understand the proofof the special termination theorem in [F13].

Example 7.4.1. We fix a lattice N = Z3. We put e1 = (1, 0, 0),e2 = (−1, 2, 0), e3 = (0, 0, 1), and e4 = (−1, 3,−3). We consider thefan

∆ = 〈e1, e3, e4〉, 〈e2, e3, e4〉, and their faces.We put X = X(∆), that is, X is the toric variety associated to the fan∆. We define torus invariant prime divisors Di = V (ei) for 1 ≤ i ≤ 4.We can easily check the following claim.

Claim. The pair (X,D1 +D3) is a Q-factorial dlt pair.

We put C = V (〈e3, e4〉) ' P1, which is a torus invariant irreduciblecurve on X. Since 〈e2, e3, e4〉 is a non-singular cone, the intersectionnumber D2 · C = 1. Therefore,

C ·D4 = −2

3

and

−(KX +D1 +D3) · C =1

3.

We note the linear relation e1 + 3e2 − 6e3 − 2e4 = 0. We put Y =X(〈e1, e2, e3, e4〉), that is, Y is the affine toric variety associated to thecone 〈e1, e2, e3, e4〉. Then we have the next claim.

Claim. The birational map f : X → Y is a flipping contractionwith respect to KX +D1 +D3 such that −D3 is f -ample.

Note that f : (X,D1 + D3) → Y is a pl flipping contraction inthe sense of [F13, Definition 4.3.1]. We note the intersection numbersC ·D1 = 1

3and D3 · C = −2. Let ϕ : X 99K X+ be the flip of f . We

254 7. SOME SUPPLEMENTARY TOPICS

note that the flip ϕ is an isomorphism around any generic points of logcanonical centers of (X,D1 +D3). We restrict the flipping diagram

Xϕ //_______

f @@@

@@@@

@ X+

f+

Y

to D3. Then we have the following diagram.

D3//_________

""EEE

EEEE

EED+

3

||xxxxxxxx

f(D3)

It is not difficult to see that D+3 → f(D3) is an isomorphism. We

put (KX + D1 + D3)|D3 = KD3 + B. Then f : D3 → f(D3) is anextremal divisorial contraction with respect to KD3 +B. We note thatB = D1|D3 .

Claim. The birational morphism f : D3 → f(D3) contracts E ' P1

to a point Q on D+3 ' f(D3) and Q is a 1

2(1, 1)-singular point on

D+3 ' f(D3). The surface D3 has a 1

3(1, 1)-singular point P , which

is the intersection of E and B. We also have the adjunction formula(KD3 +B)|B = KB + 2

3P .

Let D+i be the torus invariant prime divisor V (ei) on X+ for all i

and let B+ be the strict transform of B on D+3 .

Claim. We have

(KX+ +D+1 +D+

3 )|D+3

= KD+3

+B+

and

(KD+3

+B+)|B+ = KB+ +1

2Q.

We note that f+ : D+3 → f(D3) is an isomorphism. In particular,

D3//_________

""EEE

EEEE

EED+

3

||xxxxxxxx

f(D3)

is of type (DS) in the sense of [F13, Definition 4.2.6]. Moreover, f :B → B+ is an isomorphism but f : (B, 2

3P ) → (B+, 1

2Q) is not an

isomorphism of pairs (see [F13, Definition 4.2.5]). We note that Bis a log canonical center of (X,D1 + D3). So, we need [F13, Lemma

7.4. A SAMPLE COMPUTATION OF A LOG FLIP 255

4.2.15]. Next, we restrict the flipping diagram to D1. Then we obtainthe diagram.

D1//_________

""EEE

EEEE

EED+

1

||xxxxxxxx

f(D1)

In this case, f : D1 → f(D1) is an isomorphism.

Claim. The surfaces D1 and D+1 are smooth.

It can be directly checked. Moreover, we obtain the following ad-junction formulas.

Claim. We have

(KX +D1 +D3)|D1 = KD1 +B +2

3B′,

where B (resp. B′) comes from the intersection of D1 and D3 (resp. D4).We also obtain

(KX+ +D+1 +D+

3 )|D+1

= KD+1

+B+ +2

3B′

++

1

2F,

where B+ (resp. B′+) is the strict transform of B (resp. B′) and F isthe exceptional curve of f+ : D+

1 → f(D1).

Claim. The birational morphism f+ : D+1 → f(D1) ' D1 is the

blow-up at P = B ∩B′.

We can easily check that

KD+1

+B+ +2

3B′

++

1

2F = f+∗(KD1 +B +

2

3B′)− 1

6F.

It is obvious that KD+1

+ B+ + 23B′+ + 1

2F is f+-ample. Note that F

comes from the intersection of D+1 and D+

2 . Note that the diagram

D1//_________

""EEE

EEEE

EED+

1

||xxxxxxxx

f(D1)

is of type (SD) in the sense of [F13, Definition 4.2.6].

256 7. SOME SUPPLEMENTARY TOPICS

7.5. A non-Q-factorial flip

The author apologizes for the mistake in [F12, Example 4.4.2]. Forthe details on three-dimensional terminal toric flips, see [FSTU]. Inthis section, we explicitly construct a three-dimensional non-Q-factorialcanonical Gorenstein toric flip. We think that it is not so easy toconstruct such examples without using the toric geometry.

Example 7.5.1 (Non-Q-factorial canonical Gorenstein toric flip).We fix a lattice N = Z3. Let n be a positive integer with n ≥ 2. Wetake lattice points e0 = (0,−1, 0),

ei = (n+ 1− i,n−1∑

k=n+1−i

k, 1)

for 1 ≤ i ≤ n + 1, and en+2 = (−1, 0, 1). Note that e1 = (n, 0, 1). Weconsider the following fans.

∆X = 〈e0, e1, en+2〉, 〈e1, e2, · · · , en+1, en+2〉, and their faces,∆W = 〈e0, e1, · · · , en+1, en+2〉, and its faces, and

∆X+ = 〈e0, ei, ei+1〉, for i = 1, · · · , n+ 1, and their faces.

We define X = X(∆X), X+ = X(∆X+), and W = X(∆W ). Then wehave a diagram of toric varieties.

X //_______

ϕ AAA

AAAA

A X+

ϕ+zzzz

zzzz

W

We can easily check the following properties.

(i) X has only canonical Gorenstein singularities.(ii) X is not Q-factorial.(iii) X+ is smooth.(iv) ϕ : X → W and ϕ+ : X+ → W are small projective toric

morphisms.(v) −KX is ϕ-ample and KX+ is ϕ+-ample.(vi) ρ(X/W ) = 1 and ρ(X+/W ) = n.

Therefore, the above diagram is a desired flipping diagram. We notethat

ei + ei+2 = 2ei+1 + e0

for i = 1, · · · , n− 1 and

en + en+2 = 2en+1 +n(n− 1)

2e0.

7.5. A NON-Q-FACTORIAL FLIP 257

We recommend the reader to draw pictures of ∆X and ∆X+ whenn = 2.

By this example, we see that a flip sometimes increases the Pi-card number when the flipping variety X is not Q-factorial. Moreover,the flipped variety X+ sometimes becomes Q-factorial even when theflipping variety X is not Q-factorial.

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Index

Cdiff(X), 166D-dimension, 36D(X), 166Db

coh(X), 166Db

diff,coh(X), 167D<1, 18D=1, 18D>1, 18D≥1, 18D≤1, 18D⊥, 227D<0, 227D>0, 227D≥0, 227D≤0, 227Ddiff(X), 167KX , 24N1(X/S), 22N1(X/S), 22Z1(X/S), 22[X, ω], 14Div(X), 163Exc(f), 13Fix Λ, 114Nqklt(X, ω), 213Nqlc(X,ω), 204PerDiv(X), 162Pic(X), 22Weil(X), 162condX , 153discrep(X, ∆), 25≡, 22κ(X, D), 36κσ, 38, 145dDe, 18, 161bDc, 18, 161

Q, 14Q-Cartier, 17, 160Q-divisor, 17, 161Q-factorial, 15Q-linear equivalence, 160R, 14R-Cartier, 17, 160R-divisor, 17, 161R-linear equivalence, 160R>0, 14R≥0, 14Z, 14Z>0, 14Z≥0, 14B(D), 114B(D/S), 114CX , 153J (X, ∆), 29, 60JNLC(X, ∆), 29KX , 17K∗

X , 17QuotΦ,L

E/X/S , 165LCS(X), 213PE(X), 37QuotΦ,L

E/X/S , 165ν(X, D), 36ω-negative, 227NE(X/S)−∞, 226Mov(X/Y ), 36ρ(X/S), 22∼, 17∼Q, 17∼R, 17totaldiscrep(X, ∆), 25Weil(X)Q, 113Weil(X)R, 113

271

272 INDEX

D, 18, 161

abundance conjecture, 135abundance theorem, 149adjunction, 202, 210Alexeev’s criterion, 239ambient space, 162, 203Ambro vanishing theorem, 80ample divisors, 17Artin–Keel, 152

basepoint-free theorem, 104, 217basepoint-free theorem of

Reid–Fukuda type, 234BCHM, 112Bierstone–Milman, 165Bierstone–Vera Pacheco, 165big Q-divisor, 18big R-divisor, 19big divisor, 18birational map, 13boundary, 18, 161

canonical, 26canonical divisor, 24center, 14Chern connection, 64conductor, 153conductor ideal, 153cone singularities, 247cone theorem, 1, 104, 202, 228contractible at infinity, 227contraction theorem, 2, 227curvature form, 65

derived categories, 166discrepancy, 25divisor over X, 14divisorial contraction, 110divisorial log terminal, 30dlt blow-ups, 117DLT extension conjecture, 144double normal crossing point, 153Du Bois complexes, 165Du Bois pairs, 168dualizing complex, 14dualizing sheaf, 14

embedded, 162, 189

embedded log transformation, 185,190

Enoki injectivity theorem, 64exceptional locus, 13existence of minimal models, 113extremal, 130extremal face, 227extremal ray, 227

Fano contraction, 110finite generation of log canonical

rings, 116finiteness of marked minimal models,

113fixed divisor, 114flipping contraction, 110fractional part, 18, 161Francia’s flip, 251Fujita vanishing theorem, 68fundamental form, 65

globally embedded simple normalcrossing pairs, 203

good minimal model conjecture, 144

Hartshorne conjecture, 2Hodge theoretic injectivity theorem,

169

Iitaka dimension, 36infinitely many marked minimal

models, 119injectivity theorem for simple normal

crossing pairs, 177

kawamata log terminal, 26Kawamata–Viehweg vanishing

theorem, 45, 48, 54, 56, 59, 103,186

Kleiman’s criterion, 22Kleiman–Mori cone, 22Kodaira vanishing theorem, 43, 67,

180, 181Kodaira’s lemma, 20Kollar injectivity theorem, 63Kollar torsion-free theorem, 64Kollar vanishing theorem, 64

LCS locus, 213linear equivalence, 160

INDEX 273

log canonical, 26log canonical center, 29log canonical model, 137log canonical strata, 29log minimal model, 108log surfaces, 148log terminal, 26

minimal log canonical center, 29minimal model, 108, 111, 112, 139,

147, 149minimal model program, 112minimal model program for log

surfaces, 149minimal model program with

scaling, 115Miyaoka vanishing theorem, 61MMP, 109Mori fiber space, 110, 140, 147, 149movable cone, 36movable divisor, 36multiplier ideal sheaf, 7, 29, 60Mumford vanishing theorem, 197

Nadel vanishing theorem, 7, 61Nakayama’s numerical dimension,

38, 145nef, 22nef and log big divisor, 54, 182, 190,

206negativity lemma, 33non-klt center, 30non-klt locus, 29non-lc ideal sheaf, 29non-lc locus, 29non-qlc locus, 204non-vanishing conjecture, 144non-vanishing theorem, 104, 113Norimatsu vanishing theorem, 55normal crossing, 189normal crossing divisor, 15normal crossing pair, 189normal pairs, 120numerical dimension, 36numerical equivalence, 22numerical Iitaka dimension, 36

pairs, 14partial resolution, 164

permissible, 162, 164permissible divisor, 190pinch point, 153pl-flip, 112pl-flipping contraction, 112positive, 65principal Cartier divisor, 17projectively normal, 249pseudo-effective cone, 37pseudo-effective divisor, 37purely log terminal, 26

qlc center, 205qlc pair, 4, 205quasi-log canonical class, 204quasi-log Fano scheme, 232quasi-log pair, 204quasi-log resolution, 204quasi-log scheme, 202, 203quasi-log stratum, 204Quot scheme, 165

rational, 227rational singularities, 82rationality theorem, 104, 221real linear system, 113Reid–Fukuda vanishing theorem, 54relative big R-divisor, 19relative Hodge theoretic injectivity

theorem, 174relative quasi-log Fano scheme, 232relative quasi-log scheme, 204relative vanishing lemma, 176relatively ample at infinity, 227relatively nef, 22resolution lemma, 30round-down, 18, 161round-up, 18, 161

scheme, 13semi log canonical center, 154semi log canonical pair, 153semi log canonical stratum, 154semi-ample R-divisor, 20semi-ample divisor, 20semi-normal, 206semi-positive, 65semi-snc pair, 162Shokurov polytope, 130

274 INDEX

simple normal crossing divisor, 15,163

simple normal crossing pair, 161singularities of pairs, 26slc, 153slc center, 154slc stratum, 154small, 109Sommese’s example, 194stable augmented base locus, 113stable base locus, 113stable variety, 155star closed, 216stratum, 162, 164, 189sub kawamata log terminal, 26sub klt, 26sub lc, 26sub log canonical, 26subboundary, 18, 161support, 161supporting function, 227

Tankeev, 63terminal, 26toric polyhedron, 216total discrepancy, 25

vanishing and torsion-free theoremfor simple normal crossing pairs,178

vanishing theorem, 202, 210variety, 13Viehweg vanishing theorem, 53, 55

weak log-terminal, 32weak log-terminal singularities, 86Weil divisor, 161Whitney umbrella, 155

X-method, 105

Zariski decomposition, 62, 114

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